![Preface
In the first half of the 20th century, analysis went from studying smooth functions
to nonsmooth ones, introducing such notions as weak solutions, Sobolev spaces and
distributions. These concepts were first studied on Rn
and later on manifolds and
other smooth objects. Around 1970 came the first step towards analysis on nonsmooth
objects, when notions such as maximal functions and Lebesgue points were studied
on spaces of homogeneous type, i.e. on quasimetric spaces equipped with doubling
measures. This theory can be called zero-order analysis, as no derivatives are used.
Taking partial derivatives is not possible in metric spaces, but in the 1990s there was
a need for studying first-order analysis on nonsmooth spaces. Heinonen and Koskela
realized that upper gradients could be used as a substitute for the usual gradient. This
gave rise to Newtonian spaces, one of several attempts to define Sobolev spaces on
metric spaces, and perhaps the most fruitful one. It turned out that the potential theory
of p-harmonic functions can be extended to metric spaces through the use of upper
gradients.
The nonlinear potential theory of p-harmonic functions on Rn
has been developing
since the 1960s and has later been generalized to weighted Rn
, Riemannian manifolds,
graphs, Heisenberg groups and more general Carnot groups and Carnot–Carathéodory
spaces, and other situations. Studying potential theory of p-harmonic functions on
metric spaces generalizes and gives a unified treatment of all these cases.
There are primarily five books devoted to nonlinear potential theory, viz. Adams–
Hedberg [5], Heinonen–Kilpeläinen–Martio [171], Malý–Ziemer [258], Mizuta [287]
and Turesson [342]. They all study potential theory on unweighted ([5], [258] and
[287]) or weighted ([171] and [342]) Rn
. In [5] and [342] the focus is on higher order
potential theory, whereas the main topic in [171] and [258] is the potential theory of
p-harmonic functions (on weighted and unweighted Rn
, respectively). The main focus
in [287] is on Riesz potentials. The topics covered in our book are closest to [171], but
there are also some parts in common with [258]. There is also some overlap, especially
in Chapter 8, with the book by Giusti [146] (for unweighted Rn
).
Let us also mention the survey papers by Martio [266], which has much in common
with this book, and by Björn–Björn [47], which is based on an early version of this
book. Moreover, the forthcoming monograph Heinonen–Koskela–Shanmugalingam–
Tyson [177] has a certain overlap with the first part of this book.
This book consists of two related parts. In the first part we develop the theory of
Newtonian (Sobolev) spaces on metric spaces, and in the second part we develop the
potential theory associated with p-harmonic functions on metric spaces.
BoththeNewtonianandthe p-harmonictheoriesonmetricspaceshavenowreached
such a maturity that we think they deserve to be written in book form. So far, both
theories are scattered over a large number of different research papers published during](https://image.slidesharecdn.com/1229182-250612113446-b5066c02/85/Nonlinear-Potential-Theory-On-Metric-Spaces-Anders-Bjorn-Jana-Bjorn-9-320.jpg)
![Preface vii
The second part of the book consists of Chapters 7–14. In it we develop the potential
theory associated with p-harmonic functions. Already from the beginning we need to
use doubling, a p-Poincaré inequality and that p 1. We also assume completeness,
and these assumptions are general throughout the second part. We have taken special
care to cover the case when X is a bounded metric space.
In Chapters 7 and 8 we study the basic properties of superminimizers and show
interior regularity, whereas Chapter 9 is devoted to superharmonic functions.
In Chapter 10 we look closely at the Dirichlet problem, and in Chapter 11 on bound-
ary regularity. In Chapter 12 we consider removable singularities, which are in turn
needed to obtain the trichotomy (Theorem 13.2) motivating the study and classification
of irregular boundary points in Chapter 13.
Finally, in Chapter 14 we show that open sets can be approximated by regular sets
and give some consequences thereof, including another formulation of the Dirichlet
problem.
In Appendix A we give various examples of metric spaces satisfying our basic
assumptions (completeness, doubling and the validity of a p-Poincaré inequality). In
particular, we show that on weighted Rn
our theory coincides with the one developed in
Heinonen–Kilpeläinen–Martio [171] and other sources. When specialized to weighted
Rn
, many of the results in this book appear in [171]. We have refrained from pointing
out exactly which ones in the notes at the end of each chapter. Instead, we make [171]
as a general reference for comparison throughout the book. The reader is also referred
to the comments and references in [171]. In any case, we would like to point out that
many of the results in Chapters 10–14 do not appear in [171] (when specialized to
weighted Rn
).
In Appendix B we take a quick look at Hajłasz–Sobolev and Cheeger–Sobolev
spaces, and in Appendix C we give a short overview of the more general potential
theory of quasiminimizers.
The reader interested in open problems should observe that the item open problem
in the index gives references to all open problems stated in the book.
We started writing this book when giving a graduate course on this topic during
the autumn of 2005 in Linköping. In fact, we gave a similar course in Prague in the
autumn of 2003, and the handwritten notes we then obtained are in a sense the very
first draft of this book. We thank the participants for their active role in the courses:
Jan Kališ, Jan Malý, Petr Přecechtěl and Jiří Spurný in Prague, and Gunnar Aronsson,
Thomas Bäckdahl, Daniel Carlsson, David Färm, Thomas Karlsson, Mats Neymark,
Björn Textorius, Johan Thim and Bengt-Ove Turesson in Linköping.
Later on we have also given lecture series on these topics: Anders at the Paseky
springschool FunctionSpaces, InequalitiesandInterpolation, inJune2009, onNewto-
nian spaces, and Jana within the Taft Research Seminar at the University of Cincinnati,
during the winter term 2010. Again, we thank all the participants.
We are also grateful to Zohra Farnana, Heikki Hakkarainen, Stanislav Hencl,
David Herron, Tero Kilpeläinen, Juha Kinnunen, Riikka Korte, Visa Latvala, Tero](https://image.slidesharecdn.com/1229182-250612113446-b5066c02/85/Nonlinear-Potential-Theory-On-Metric-Spaces-Anders-Bjorn-Jana-Bjorn-11-320.jpg)
![Chapter 1
Newtonian spaces
Throughout the book we will study potential theory on the metric space X. We make
some general assumptions on X in the beginning of Section 1.1 below. These as-
sumptions will hold throughout the book. In many of the chapters we will make some
additional assumptions holding throughout that chapter. These assumptions will al-
ways be listed at the very beginning of the chapter. Occasionally we will state some
additional assumptions at the very beginning of a section.
In the first half of this book, Chapters 1–6, we introduce and study Sobolev type
spaces on metric measure spaces. These spaces, called Newtonian spaces, will be the
natural setting for p-harmonic functions and potential theory in the second half of the
book. In Appendix B we mention some other possible definitions of Sobolev type
spaces on metric spaces and relate them to our definition.
Let us first take a very quick look at first-order Sobolev spaces on Rn
. For 1
p 1 and f 2 Lp
.Rn
/ we define
kf kp
W 1;p.Rn/
D
Z
Rn
.jf jp
C jrf jp
/ dx;
where rf is the weak or distributional gradient of f . The Sobolev space W 1;p
.Rn
/
is then given by
W 1;p
.Rn
/ D ff W kf kW 1;p.Rn/ 1g:
To define W 1;p
.Rn
/, one uses the gradient, i.e. the directional derivatives. In metric
spaces we cannot talk about directions, so that is problematic. However, we do not
really use the vector rf , only the scalar jrf j is necessary to define W 1;p
.Rn
/, and
for jrf j there is a possible counterpart in metric spaces, called upper gradient and
introduced by Heinonen–Koskela [173], [174].
Before defining upper gradients, in Section 1.3, we introduce our metric spaces and
give some background results. Let us mention here that in Section 1.7 we give a short
overview of some examples of metric spaces on which the theory of Newtonian spaces
is useful. A more detailed discussion is given in Appendix A.
1.1 The metric space X and some notation
We assume throughout the book that 1 p 1 and that X D .X; d; / is a metric
space endowed with a metric d and a positive complete Borel measure such that
.B/ 1 for all balls B X
(we make the convention that balls are nonempty and open).](https://image.slidesharecdn.com/1229182-250612113446-b5066c02/85/Nonlinear-Potential-Theory-On-Metric-Spaces-Anders-Bjorn-Jana-Bjorn-17-320.jpg)
![2 1 Newtonian spaces
We emphasize that to start with we do not assume that X is complete nor locally
compact. We will however assume that X contains at least two points (to avoid special
formulations needed to cover the pathological cases when X is empty and when X is
a singleton).
We also emphasize that the -algebra on which is defined is obtained by the
completion of the Borel -algebra. We further extend as an outer measure on X, so
that for an arbitrary set A X we have
.A/ D inff.E/ W E A is a Borel setg:
Proposition 1.1. The measure is Borel regular, i.e. for every A X there is a Borel
set E A such that .A/ D .E/.
Note that this is the definition of Borel regularity used by Federer [123], Sec-
tion 2.2.3, Mattila [275], Definition 1.5, and Heinonen [169], p. 3, whereas Rudin [311]
has a more restrictive definition of Borel regularity which is not always fulfilled for our
spaces.
Proof. If .A/ D 1, then we may choose E D X. Otherwise we can find Borel sets
Ej A with .Ej / .A/ C 1=j , j D 1; 2; ::: . Letting E D
T1
jD1 Ej completes
the proof.
Let us here record some notation used throughout the book. We use the following
notation for balls
B.x0; r/ WD fx 2 X W d.x; x0/ rg;
and for B D B.x0; r/ and 0, we let B D B.x0; r/. Further, we always assume
that , 0
, 00
and t , t 2 R, are nonempty open subsets of X.
In the metric space X, a.e. always means almost everywhere with respect to the
given measure , unless otherwise stated.
For a measurable set A X with 0 .A/ 1, we define the mean-value
integral (or integral average) of f over A as
«
A
f d WD
1
.A/
Z
A
f d;
whenever the right-hand side exists, which it does, in particular, if f 2 L1
.A/ or if f
is nonnegative and measurable on A. We shall also use the abbreviation
fA WD
«
A
f d:
In general we allow semicontinuous functions to be extended real-valued, i.e. to
take values in x
R D Œ1; 1. On the other hand, continuous functions are always
assumed to be real-valued, unless mentioned otherwise.](https://image.slidesharecdn.com/1229182-250612113446-b5066c02/85/Nonlinear-Potential-Theory-On-Metric-Spaces-Anders-Bjorn-Jana-Bjorn-18-320.jpg)
![1.2 Preliminaries 5
Proof. Write G as a union of its components. Each component is necessarily an open
interval. Associate with each component one of its rational points. Then we see that
the set of components cannot have larger cardinality than Q.
Proposition 1.5. Let Y be a metric space. Then the following are equivalent:
(a) Y is separable;
(b) Y is second countable, i.e. it has a countable open base;
(c) Y is Lindelöf, i.e. every open cover of Y has a countable subcover.
It is easily seen that these properties are also inherited by any subset of Y (in the
induced topology).
This is well known, see e.g. Exercise 5, p. 194, in Munkres [296], or Figure 10
in Steen–Seebach [327]. However, we have not found one good reference with full
proofs of all implications.
Proof. (a) , (b) See Theorem 2, p. 177, in Kuratowski [235].
(b) ) (c) See Theorem 2, p. 176, in [235].
(c) ) (b) See the second paragraph on p. 35 in Steen–Seebach [327].
In our considerations on the metric measure space X, the support of will be of
main interest. The following result is therefore useful. See also Proposition 1.53 and
the comment before it.
Note, however, that we do not want to assume that X D supp in general, since
we want to be able to consider arbitrary subsets of X as metric spaces on their own and
it then may not be true that the support of the restriction of the underlying measure is
the whole subset under consideration.
Recall that x 2 supp , the support of , if and only if .B.x; r// 0 for every
r 0. Or in other terms X n supp is the union of all balls with zero measure, and is
the largest open set with zero measure, in the sense that it contains any open set with
zero measure. (The support of a function is similarly defined as the complement of the
largest open set on which f is 0.)
Proposition 1.6. The metric space supp is separable, second countable and Lindelöf.
Before proving Proposition 1.6, let us make some remarks. The following simple
but very useful covering lemma is well known.
Lemma 1.7 (5-covering lemma). Let B be a family of balls in X with uniformly
bounded radii. Then there exists a subfamily B0
B of pairwise disjoint balls such
that [
B2B
B
[
B2B0
5B:
If X is separable, then the subfamily B0
can be chosen to be countable.](https://image.slidesharecdn.com/1229182-250612113446-b5066c02/85/Nonlinear-Potential-Theory-On-Metric-Spaces-Anders-Bjorn-Jana-Bjorn-21-320.jpg)
![6 1 Newtonian spaces
Proof. For a proof of the first part see e.g. Ambrosio–Tilli [17], Theorem 2.2.3, Hei-
nonen [169], Theorem 1.2, or Federer [123], Theorem 2.8.4 (choose D 2 and let ı
be the radius of the balls in Federer’s formulation).
For the second part assume that X is separable. Then X is Lindelöf by Proposi-
tion 1.5, and hence we can find a countable subfamily B00
B0
such that
S
B2B00 5B D
S
B2B0 5B.
Remark 1.8. The constant 5 in the 5-covering lemma is not optimal, but has become
standard as one wants to have as simple an argument as possible. With just a slight extra
thought most proofs actually show that one may replace 5 by 3 C for any 0. In
Federer [123], Theorem 2.8.4, this is even a consequence of the statement, just choose
D 1 C 1
2
. The following example shows that one cannot replace 5 by 3.
Example 1.9. Let X D R and B2j1 D B.22=j; 21=j/ for j D 1; 2; ::: . In other
terms, B2j1 is the interval .1=j; 4 3=j/. Let also B2j D B2j1, j D 1; 2; ::: .
Then 0 2 Bj for all j D 1; 2; ::: , and thus any disjoint subfamily must consist of just
one ball, say Bk. If k D 2j 1 is odd, then 3Bk is the interval .4 C 1=j; 8 5=j/
and therefore does not cover
S1
jD1 Bj D .4; 4/. The case when k is even is similar,
and hence the constant 5 in the 5-covering lemma cannot be replaced by 3.
Proof of Proposition 1.6. Fix x0 2 Y WD supp and let 0 and R 0. Let
B D fB.x; / W x 2 Y B.x0; R/g:
By the 5-covering lemma (Lemma 1.7), we can find a subfamily B0
B of pairwise
disjoint balls so that
Y B.x0; R/
[
B2B
B
[
B2B0
5B:
As X
B2B0
.B/ D
[
B2B0
B
.B.x0; R C // 1;
and .B/ 0 for every B 2 B0
, we conclude that B0
is a countable set. The centres
of the balls B 2 B0
make a countable -net in Y B.x0; R/. Letting D 1=n,
n D 1; 2; ::: , shows that Y B.x0; R/ is separable. Thus Y D
S1
jD1.Y B.x0; j//
is also separable.
That Y is second countable and Lindelöf follows from Proposition 1.5.
Lemma 1.10 (Cavalieri’s principle). Let be a measure on X. If f W X ! Œ0; 1 is
a -measurable function and 0 q 1, then
Z
X
f q
d D q
Z 1
0
q1
.fx W f .x/ g/ d:](https://image.slidesharecdn.com/1229182-250612113446-b5066c02/85/Nonlinear-Potential-Theory-On-Metric-Spaces-Anders-Bjorn-Jana-Bjorn-22-320.jpg)
![1.2 Preliminaries 7
The Cavalieri principle is a standard tool in integration theory, we state it here for
the reader’s convenience. Sometimes it is the case q D 1 that is called the Cavalieri
principle, but the general case follows after the change of variable 7! q
.
The Cavalieri principle can be proved in several different ways, e.g. using the Fubini
theorem. The most convenient way actually depends on how the Lebesgue integral is
defined and the proof truly belongs to integration theory. We refer the reader in need
to Folland [124], Proposition 6.24.
Another result which eventually will be needed is that continuous functions can
be approximated by Lipschitz functions on compact sets. Recall that f W Y ! R is
Lipschitz on the metric space .Y; d/ if there exists a constant C such that
jf .x/ f .y/j Cd.x; y/ for all x; y 2 Y:
Proposition 1.11. Let .Y; d/ be a compact metric space and let u 2 C.Y / and 0.
Then there is v 2 Lip.Y / such that supy2Y ju.y/ v.y/j .
This follows from the Stone–Weierstrass theorem (see e.g. Rudin [312], p. 122),
but for the reader’s convenience, we give an elementary proof of this fact.
Proof. We can assume that minY u D 0 and let Gk D fx 2 Y W u.x/ kg for
positive integers k. Clearly, only finitely many Gk are nonempty. As Y is compact,
so is each x
Gk, and hence there exists ı 0 so that dist.Gk; Y n Gk1/ ı whenever
Gk ¤ ¿. Let, for x 2 Y ,
u.x/ D
X
k1
1
dist.Gk; x/
ı
C
;
where we only sum over nonempty Gk. Then u 2 Lip.Y / (with constant =ı) and
ju.x/ u.x/j for all x 2 Y .
We will also need to approximate lower semicontinuous functions by Lipschitz
functions from below.
Proposition 1.12. Let .Y; d/ be a compact metric space and let uW Y ! .1; 1 be
lower semicontinuous. Then there exists an increasing sequence fuj g1
jD1 of Lipschitz
functions such that uj .x/ ! u.x/, as j ! 1, for all x 2 Y .
If u 0, then we may choose uj 0, j D 1; 2; :::.
In fact it is fairly easy to see that the uj constructed below is the largest j -Lipschitz
function u (apart from the case when u 1 when no such largest function exists).
Proof. As u is lower semicontinuous on a compact set, it attains its minimum, which
by assumption is not 1. Hence, without loss of generality, we may assume that
u 0. If u 1, then we can take uj j, j D 1; 2; ::: , and we therefore exclude
this case below. Fix a positive integer j, and let
uj .x/ D inf
y2Y
.u.y/ C jd.x; y//:](https://image.slidesharecdn.com/1229182-250612113446-b5066c02/85/Nonlinear-Potential-Theory-On-Metric-Spaces-Anders-Bjorn-Jana-Bjorn-23-320.jpg)
![8 1 Newtonian spaces
It is easy to see that uj is real-valued and that 0 uj ujC1 u.
Let us first show that uj is j-Lipschitz. Let x; y 2 Y and 0. Then we can find
z 2 Y such that uj .x/ u.z/ C jd.z; x/ . It follows that
uj .y/ u.z/ C jd.z; y/ u.z/ C jd.z; x/ C jd.x; y/ uj .x/ C jd.x; y/ C :
Letting ! 0 shows that uj .y/ uj .x/ C jd.x; y/. As we also have uj .x/
uj .y/ C jd.x; y/, we have shown that uj is j-Lipschitz.
Let x 2 Y be arbitrary. It remains to show that uj .x/ ! u.x/, as j ! 1. Let
u.x/. As u is lower semicontinuous we can find a ball B D B.x; r/ in Y such
that u in B. For j =r we have
uj .x/ D min
inf
y2B
.u.y/ C jd.x; y//; inf
y2Y nB
.u.y/ C jd.x; y//
min
inf
y2B
; inf
y2Y nB
jr
D :
Hence uj .x/ ! u.x/, as j ! 1.
1.3 Upper gradients and the Newtonian space N1;p
In this section we introduce upper gradients as a substitute for the modulus of the
usual gradient. We will later see that under some assumptions it is possible to control
a function by its upper gradients. On the other hand, in situations as in Example 1.22
the upper gradients do not give any information about the function. To begin with, we
will leave this question aside and just concentrate on developing the theory of upper
gradients, and p-weak upper gradients, whatever use they may have. In Chapters 3
and 4 we will introduce some conditions that will be sufficient for the upper gradients
to control their functions.
By a curve in X we will mean a rectifiable nonconstant continuous mapping from
a compact interval. (For us only such curves will be interesting, in general a curve is a
continuous mapping from an interval. A rectifiable curve is a curve with finite length.)
A curve can thus be parameterized by arc length ds, and we will always assume that
all curves are parameterized by arc length, see e.g. Ambrosio–Tilli [17], Section 4.2,
or Heinonen [169], Section 7.1. Note that every curve is Lipschitz continuous with
respect to its arc length parameterization. We let .X/ denote the family of all curves
on X. By abuse of notation we also denote the image of a curve by .
Definition 1.13. A nonnegative Borel function g on X is an upper gradient of an
extended real-valued function f on X if for all curves W Œ0; l ! X,
jf . .0// f . .l //j
Z
g ds: (1.1)](https://image.slidesharecdn.com/1229182-250612113446-b5066c02/85/Nonlinear-Potential-Theory-On-Metric-Spaces-Anders-Bjorn-Jana-Bjorn-24-320.jpg)
![1.5 p-weak upper gradients and modulus of curve families 21
Corollary 1.44. Let g be a p-weak upper gradient of f , and let Q
g D g a.e., Q
g 0.
Then Q
g is also a p-weak upper gradient of f . In particular, there is a Borel p-weak
upper gradient g0
of f such that g0
D g a.e.
The following example shows that the corresponding result for upper gradients is
false.
Example 1.45. Let f W R2
! R be given by f ..x1; x2// D x1. Then g D 1 is an
upper gradient of f . Let further g0
D gR2nR. Then g0
D g a.e. and g0
is a nonnegative
Borel function which is not an upper gradient of f . Nevertheless, g0
is a p-weak upper
gradient of f , by Corollary 1.44.
The following result shows that the N 1;p
-norm is not changed if the infimum in
Definition 1.17 is taken over all p-weak upper gradients of u. Note that even though
this result is mainly of interest when g 2 Lp
.X/, this is not required.
Lemma 1.46. Let g be a p-weak upper gradient of f . Then there exist upper gradients
gj so that
lim
j!1
kgj gkLp.X/ D 0: (1.6)
Proof. First we can find a nonnegative Borel function g0
such that g0
D g a.e. By
Lemma 1.43, g0
is also a p-weak upper gradient of f . Let consist of those curves
W Œ0; l ! X such that
jf . .0// f . .l //j 6
Z
g0
ds:
By assumption, Modp. / D 0, and hence by Proposition 1.37, there is a nonnegative
Borel function 2 Lp
.X/ such that
R
ds D 1 for all 2 . Let finally gj D
g0
C =j . Then gj is an upper gradient of f and (1.6) holds.
The following result shows that p-weak upper gradients are a natural generalization
of the (modulus of the) usual gradients. It is an improvement of Corollary 1.15 and
will be further refined in Propositions A.3 and A.11.
Corollary 1.47. If X D Rn
and f W Rn
! R is locally Lipschitz, then jrf j (or more
precisely any of its nonnegative everywhere defined representatives) is a p-weak upper
gradient of f .
Proof. Proposition 1.14 shows that lip f is an upper gradient of f . By the Rademacher
theorem (see e.g. Theorem 2.2.1 in Ziemer [361]), f is differentiable at a.e. x 2 Rn
.
For such x and all y 2 B.x; r/ we have
jf .y/ f .x/j D jrf .x/ .y x/ C o.jy xj/j r jrf .x/j C o.r/
and hence lip f .x/ D jrf .x/j. The result now follows from Corollary 1.44.](https://image.slidesharecdn.com/1229182-250612113446-b5066c02/85/Nonlinear-Potential-Theory-On-Metric-Spaces-Anders-Bjorn-Jana-Bjorn-37-320.jpg)
![24 1 Newtonian spaces
Recall our convention that 1 1 D 1. In particular, the second alternative
in (1.7) does not hold for jŒ0;t if t D 0 and u. .0// D ˙1, and therefore the
cases t D 0 and t D l needed separate treatment above. Note that in the context of
Corollary 1.51 this is not necessary.
Lemma 1.28 has a direct counterpart for p-weak upper gradients. (Assuming that
gi are merely p-weak upper gradients, we obtain a p-weak upper gradient g with the
same proof.) However, using Proposition 1.50 (with E D fx 2 X W u.x/ D 1g), we
can show the following stronger result.
Lemma 1.52. Letuj befunctionswithp-weakuppergradientsgj , andletu D supj uj ,
g D supj gj and E D fx W u.x/ D 1g. If .E/ D 0, then g is a p-weak upper
gradient of u.
Note that unlike in Lemma 1.28, here we do not assume that uj are uniformly
bounded from above. Recall also the discussion in Remark 1.29 showing that without
any condition on E this lemma would be false.
Proof. Let j be the set of the exceptional curves along which gj is not an upper
gradient of uj , j D 1; 2; ::: . Let W Œ0; l ! X be a curve not in
S1
jD1 j and let
x D .0/ and y D .l /. Assume that either u.x/ 1 or u.y/ 1. Then, as in
the proof of Lemma 1.28,
ju.x/ u.y/j sup
j
juj .x/ uj .y/j sup
j
Z
gj ds
Z
g ds:
It follows from Corollary 1.51 that g is a p-weak upper gradient of u.
Finally, we show that for almost all purposes we can forget about X n supp and
replace X by supp . Note, however, that we do not want to assume X D supp in
general, see the comment before Proposition 1.6.
Proposition 1.53. Let Z D X n supp . Then Cp.Z/ D 0 and Modp. Z/ D 0.
Proof. As Z is open, the function g D 1Z is an upper gradient of u D Z as
R
g ds D 1 for any curve starting in Z and ending in supp . Hence Cp.Z/
kukp
N1;p.X/
D 0. That also Modp. Z/ D 0 follows from Proposition 1.48.
1.6 Banach space and ACCp
On Rn
it is well known that every Sobolev function has a representative which is
ACL, i.e. absolutely continuous on almost every line parallel to the axes, see e.g.
Theorem 2.1.4 in Ziemer [361]. On metric spaces we have no preferred lines, but we
can obtain a stronger result, Theorem 1.56 below. In fact, for this we only need the
existence of a p-integrable upper gradient, which justifies the following definition.](https://image.slidesharecdn.com/1229182-250612113446-b5066c02/85/Nonlinear-Potential-Theory-On-Metric-Spaces-Anders-Bjorn-Jana-Bjorn-40-320.jpg)
![1.6 Banach space and ACCp 25
Definition 1.54. We say that a measurable function belongs to the Dirichlet space
Dp
.X/ if it has an upper gradient in Lp
.X/.
In view of Lemma 1.46 it is equivalent to assume the existence of a p-weak upper
gradient in Lp
.X/. Clearly, N 1;p
.X/ Dp
.X/, and Lemma 1.16 immediately
implies that Dp
.X/ is a vector space.
Theorem 1.55. The space Dp
.X/ is a lattice.
Proof. The proof of Theorem 1.20 in fact also yields this result.
Theorem 1.56. If u 2 Dp
.X/, then u 2 ACCp.X/, i.e. u is absolutely continuous on
p-a.e. curve in the sense that u ı W Œ0; l ! R is absolutely continuous for p-a.e.
curve in X.
Let us recall the definition of absolute continuity.
Definition 1.57. A function f W Œa; b ! R is absolutely continuous on Œa; b if for
every 0 there is ı 0 such that
n
X
iD1
jf .bi / f .ai /j
for any n and any a a1 b1 a2 b2 an bn b such that
n
X
iD1
.bi ai / ı:
This is not the right place to study the basic facts about absolutely continuous
functions. Let us however recall that a large motivation for their study is the fact that
f is absolutely continuous on Œa; b if and only if f 0
2 L1
.Œa; b/ and
f .x/ D f .a/ C
Z x
a
f 0
.t/ dt for all x 2 Œa; b:
See e.g. Folland [124], Theorem 3.35, or Rudin [311], Theorem 7.20 together with the
comments just before Definition 7.17.
We will need the following simple facts.
Lemma 1.58. If u and v are absolutely continuous on Œa; b, then u C v, maxfu; vg
and uv are also absolutely continuous on Œa; b.
If uW Œa; b ! Œc; d is absolutely continuous and v W Œc; d ! R is Lipschitz, then
v ı u is also absolutely continuous on Œa; b.](https://image.slidesharecdn.com/1229182-250612113446-b5066c02/85/Nonlinear-Potential-Theory-On-Metric-Spaces-Anders-Bjorn-Jana-Bjorn-41-320.jpg)
![34 1 Newtonian spaces
1.8 Notes
Upper gradients were introduced under the name very weak gradients by Heinonen
and Koskela in [173], [174]. They were renamed upper gradients, for obvious reasons,
when Koskela and MacManus [229] studied p-weak upper gradients for the first time.
In Semmes [314], Definition 1.8, upper gradients are called generalized gradients.
Cheeger [91] defines generalized upper gradients, which for p 1 are roughly the
same as p-weak upper gradients. Cheeger also defines Sobolev spaces on general
metric spaces that coincide with y
N 1;p
.X/ if p 1, see Appendix B.2.
In some of the early papers on upper and p-weak upper gradients, it is not stated
explicitly if the (p-weak) upper gradients are required to be Borel functions or not.
However, since Borelness is sometimes used it may have been implicitly assumed.
Requiring that upper gradients be Borel functions has some definite advantages. First
of all, the concept of upper gradients becomes totally independent of the measure, and
second, the curve integrals involved are always defined.
As for p-weak upper gradients the situation is different. At first (when care was
taken) it was assumed that p-weak upper gradients be Borel, the reasons for this being
that it would prevent any problems with the curve integrals not being defined, and in
view of Proposition 1.2 it was not believed to do any harm. However, when working
on the book Heinonen–Koskela–Shanmugalingam–Tyson [177], it was observed by
Heinonen (as communicated to us by Shanmugalingam [322]), that care has to be
taken in glueing formulas such as in Section 2.4, as the glued p-weak upper gradient
may fail to be Borel even if the initial p-weak upper gradients are Borel. The immediate
response was that one has to take extra care when formulating similar results.
In Björn–Björn [44], Section 3 (which incidentally does not appear in Björn–
Björn [45]), a different remedy was suggested which we follow here: to omit the
assumption of Borelness for p-weak upper gradients. Let us name some of the rea-
sons for this: (a) In Definition 1.32 it is enough if the curve integrals are defined for
p-a.e. curve, and in fact this is automatic for nonnegative measurable functions, by
Lemma 1.43. (b) Just requiring measurability of course means that the concept of
p-weak upper gradients, in contrast to that of upper gradients, is dependent on the
measure. However, even the concept of Borel p-weak upper gradients is dependent on
the measure as the inequality is required to just hold for p-a.e. curve. (c) The glueing
formulas in Section 2.4 become more appealing.
Also, how to interpret j1 1j is often not mentioned explicitly. However, tak-
ing into account Corollary 1.51, it makes no difference for p-weak upper gradients
how j1 1j is interpreted, at least not when only Newtonian functions are under
consideration.
Lemma 1.34 for metric spaces is in Heinonen–Koskela [174], p. 9, where they
observe that the proofs of the corresponding Euclidean results in Fuglede [128] carry
over verbatim. Fuglede was the first one to study p-modulus. The 2-modulus and
its inverse extremal length were introduced by Beurling in the early 1930s, but first
published in the joint papersAhlfors–Beurling [6], [7]. Proposition 1.37 also goes back](https://image.slidesharecdn.com/1229182-250612113446-b5066c02/85/Nonlinear-Potential-Theory-On-Metric-Spaces-Anders-Bjorn-Jana-Bjorn-50-320.jpg)
![1.8 Notes 35
to Fuglede [128]. A thorough study of extremal length and p-modulus for p 1 in
weighted Rn
is in Ohtsuka [299].
Newtonian spaces were first defined in Shanmugalingam [318] and [319]. Therein
one can find Theorems 1.20 and 1.71, Corollaries 1.21 and 1.72, and Lemma 1.42.
Also Theorem 1.56 and Proposition 1.59 (stated only for u 2 N1;p
.X/) can be found
in [319], as well as the “only if” part of Proposition 1.48. The “if” part seems to
be new here. An equivalence between the modulus and a variational capacity (see
Section 6.3) has been obtained by Heinonen–Koskela [174] and Kallunki [Rogovin]–
Shanmugalingam [194]. Some estimates for the modulus of curve families in ring
domains on metric spaces can be found inAdamowicz–Shanmugalingam [3] and Garo-
falo–Marola [138].
Lemma 1.46 was first proved by Koskela–MacManus [229], Lemma 2.4. Lem-
ma 1.43 is from Björn–Björn [44] (but is not included in [45]), while Corollary 1.51
is from Björn–Björn–Parviainen [54]. The direction in Proposition 1.63 which is not
covered by Theorem 1.56, as well as Propositions 1.66 and 1.67, seems to be new here.
If X is a complete doubling p-Poincaré space, then any function with an upper
gradient in Lp
.X/ is measurable, byTheorem 4.52. Thus the measurability assumption
can be dropped from the definition of the Dirichlet space Dp
.X/, Definition 1.54,
under these assumptions. In general, however, it is easy to see that measurability is
not a redundant assumption in Definition 1.54, consider e.g. the von Koch snowflake
curve in Example 1.23.
For p D 1, one often studies BV-functions, functions of bounded variation, which
in some problems appear as the natural limit of W 1;p
, as p ! 1C. We have not
pursued that, since our main interest is to study the theory for p 1 and we in-
clude the case p D 1 only when it comes for free. BV-functions have been studied
on metric spaces by Ambrosio [14], [15], Camfield [83], Miranda [286], Ambrosio–
Miranda–Pallara [16], Kinnunen–Korte–Shanmugalingam–Tuominen [210], [211],
Hakkarainen–Kinnunen [156] and Hakkarainen–Shanmugalingam [157].
A further generalization of Newtonian spaces are Orlicz–Sobolev spaces based
on upper gradients. They have been studied by e.g. Aïssaoui [12], [13], Heikkinen–
Tuominen [167], Mocanu [291] and Tuominen [339], [340], [341]. Estimates and in-
equalities for Orlicz–Sobolev capacities on metric spaces can be found inAïssaoui [11],
J. Björn–Onninen [71], Costea [102] and Tuominen [339]. Mocanu [288], [289], [290]
studies Newtonian spaces and capacities in even more general settings. Variable ex-
ponent Newtonian spaces N1;p.x/
were studied by Harjulehto–Hästö–Pere [160] and
Futamura–Harjulehto–Hästö–Mizuta–Shimomura [129].
Throughout the book we have restricted ourselves to real-valued Newtonian func-
tions. Newtonian spaces for Banach-space valued functions were studied by Heino-
nen–Koskela–Shanmugalingam–Tyson [176], [177].
The (first part of the) 5-covering lemma (Lemma 1.7) has many names in the
literature, including the basic covering lemma and the simple Vitali lemma and is
often attributed to Vitali [348] or Wiener [353]. However, Vitali [348] obtains a much
stronger result under stronger assumptions: he requires every point in the set E under](https://image.slidesharecdn.com/1229182-250612113446-b5066c02/85/Nonlinear-Potential-Theory-On-Metric-Spaces-Anders-Bjorn-Jana-Bjorn-51-320.jpg)
![36 1 Newtonian spaces
consideration to be covered by arbitrarily small balls (a so-called Vitali covering) and
obtains a disjoint covering of all of E but for a set of measure zero. (Vitali’s formulation
is a bit different, but it is relatively easy to see that it is equivalent to the modern
formulation, especially for open E.) It is therefore more correct to reserve the name
Vitali covering theorem for this result.
Wiener [353], on the other hand, may have been the first to consider coverings
which need not be arbitrarily small around every point. He assumes that every point
in E is the centre of some ball in the covering and finds a disjoint subfamily whose
measure is at least a fixed positive portion of the measure of E. However, he does not
consider blow-ups of the disjoint balls (not even in the proof).
Morse [292], Theorem 3.5, seems to be the first one proving the 5-covering lemma
as formulated here, and he does so in metric spaces. In fact, Morse studies coverings
by more general sets and the 5-covering lemma is a special case of his result. This may
have been the first time when the conclusion is that certain blow-ups of the disjoint sets
(balls in our case) cover E, rather than a measure-theoretic conclusion as in Vitali’s
and Wiener’s theorems. However, already Banach [27], when giving a new proof of
(the modern formulation of) Vitali’s covering theorem, shows that certain blow-ups of
the disjoint balls cover E, but he does not obtain the 5-covering lemma since he has
stronger assumptions. He too studies coverings by more general sets than balls.
Federer [123], Theorem 2.8.4, proves an even more general result than Morse, from
which the 5-covering lemma follows by choosing D 2 and letting ı be the radius of
the balls. Choosing D 1 C 1
2
gives a .3 C /-covering lemma, cf. Remark 1.8.
Let us finally mention that the name Newtonian comes from the fact that the def-
inition of upper gradients is related to the Newton–Leibniz formula, and the notation
L1;p
was already reserved for other spaces.](https://image.slidesharecdn.com/1229182-250612113446-b5066c02/85/Nonlinear-Potential-Theory-On-Metric-Spaces-Anders-Bjorn-Jana-Bjorn-52-320.jpg)
![BELLA'S BUREAU.[4]
A STORY IN THREE SCARES.
SCARE THE FIRST.
I almost flung myself into Dick Vandeleur's arms when he entered
my library that evening.
Can you imagine why I sent for you in such a deuce of a hurry? I
blurted out, embracing him effusively in my pleasure at seeing him.
Well, I did think there might have been a woman in the case, he
drawled, in his deliberate way, stopping to adjust his neck-tie, which
had worked its way over his ear during the struggle. But then, as I
happened to have acted as your best man only two months ago,
when you married the most charming of women, why, b'Jove, I—
Well, it is a woman, I groaned, cutting his speech short.
The devil!
Yes, and the very worst kind, I fancy, if thoroughly aroused.
But, my deah boy, with such a wife it's—it's—it's—
Yes, it's all that and a good deal more, I growled, gloomily. Don't
add to my misery with your ill-timed reproaches. Richard, a back
number of my unsavory career has turned up to deprive me of my
appetite and blight my being. You remember Bella Bracebridge, of
the nimble toes, at whose shrine I worshipped so long and so
idiotically? Well, I received a letter from her only yesterday.
No!—incredulously.](https://image.slidesharecdn.com/1229182-250612113446-b5066c02/85/Nonlinear-Potential-Theory-On-Metric-Spaces-Anders-Bjorn-Jana-Bjorn-67-320.jpg)
Nonlinear Potential Theory On Metric Spaces Anders Bjorn Jana Bjorn
- 1. Nonlinear Potential Theory On Metric Spaces Anders Bjorn Jana Bjorn download https://ebookbell.com/product/nonlinear-potential-theory-on- metric-spaces-anders-bjorn-jana-bjorn-2458364 Explore and download more ebooks at ebookbell.com
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- 5. EMS Tracts in Mathematics 17
- 6. EMS Tracts in Mathematics Editorial Board: Carlos E. Kenig (The University of Chicago, USA) Andrew Ranicki (The University of Edinburgh, Great Britain) Michael Röckner (Universität Bielefeld, Germany, and Purdue University, USA) Vladimir Turaev (Indiana University, Bloomington, USA) Alexander Varchenko (The University of North Carolina at Chapel Hill, USA) This series includes advanced texts and monographs covering all fields in pure and applied mathematics. Tracts will give a reliable introduction and reference to special fields of current research. The books in the series will in most cases be authored monographs, although edited volumes may be published if appropriate. They are addressed to graduate students seeking access to research topics as well as to the experts in the field working at the frontier of research. 1 Panagiota Daskalopoulos and Carlos E. Kenig, Degenerate Diffusions 2 Karl H. Hofmann and Sidney A. Morris, The Lie Theory of Connected Pro-Lie Groups 3 Ralf Meyer, Local and Analytic Cyclic Homology 4 Gohar Harutyunyan and B.-Wolfgang Schulze, Elliptic Mixed, Transmission and Singular Crack Problems 5 Gennadiy Feldman, Functional Equations and Characterization Problems on Locally Compact Abelian Groups 6 Erich Novak and Henryk Woz , niakowski, Tractability of Multivariate Problems. Volume I: Linear Information 7 Hans Triebel, Function Spaces and Wavelets on Domains 8 Sergio Albeverio et al., The Statistical Mechanics of Quantum Lattice Systems 9 Gebhard Böckle and Richard Pink, Cohomological Theory of Crystals over Function Fields 10 Vladimir Turaev, Homotopy Quantum Field Theory 11 Hans Triebel, Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration 12 Erich Novak and Henryk Woz , niakowski, Tractability of Multivariate Problems. Volume II: Standard Information for Functionals 13 Laurent Bessières et al., Geometrisation of 3-Manifolds 14 Steffen Börm, Efficient Numerical Methods for Non-local Operators. 2-Matrix Compression, Algorithms and Analysis 15 Ronald Brown, Philip J. Higgins and Rafael Sivera, Nonabelian Algebraic Topology. Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids 16 Marek Janicki and Peter Pflug, Separately Analytical Functions
- 7. Anders Björn Jana Björn Nonlinear Potential Theory on Metric Spaces
- 8. Authors: Anders Björn Department of Mathematics Linköpings universitet SE-581 83 Linköping Sweden E-mail: anbjo@mai.liu.se 2010 Mathematical Subject Classification: 31-02, 31E05; 28A12, 30L99, 31C05, 31C15, 31C40, 31C45, 35B45, 35B65, 35D30, 35J20, 35J25, 35J60, 35J67, 35J70, 35J92, 46E35, 47J20, 49J10, 49J27, 49J40, 49J52, 49N60, 49Q20, 58C99, 58J05, 58J32 Key words: Boundary regularity, capacity, Dirichlet problem, doubling measure, interior regularity, metric space, minimizer, Newtonian space, nonlinear, obstacle problem, Perron solution, p-harmonic function, Poincaré inequality, potential theory, Sobolev space, upper gradient ISBN 978-3-03719-099-9 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broad- casting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 2011 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: info@ems-ph.org Homepage: www.ems-ph.org Typeset using the author’s TEX files: I. Zimmermann, Freiburg Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 9 8 7 6 5 4 3 2 1 Jana Björn Department of Mathematics Linköpings universitet SE-581 83 Linköping Sweden E-mail: jabjo@mai.liu.se
- 9. Preface In the first half of the 20th century, analysis went from studying smooth functions to nonsmooth ones, introducing such notions as weak solutions, Sobolev spaces and distributions. These concepts were first studied on Rn and later on manifolds and other smooth objects. Around 1970 came the first step towards analysis on nonsmooth objects, when notions such as maximal functions and Lebesgue points were studied on spaces of homogeneous type, i.e. on quasimetric spaces equipped with doubling measures. This theory can be called zero-order analysis, as no derivatives are used. Taking partial derivatives is not possible in metric spaces, but in the 1990s there was a need for studying first-order analysis on nonsmooth spaces. Heinonen and Koskela realized that upper gradients could be used as a substitute for the usual gradient. This gave rise to Newtonian spaces, one of several attempts to define Sobolev spaces on metric spaces, and perhaps the most fruitful one. It turned out that the potential theory of p-harmonic functions can be extended to metric spaces through the use of upper gradients. The nonlinear potential theory of p-harmonic functions on Rn has been developing since the 1960s and has later been generalized to weighted Rn , Riemannian manifolds, graphs, Heisenberg groups and more general Carnot groups and Carnot–Carathéodory spaces, and other situations. Studying potential theory of p-harmonic functions on metric spaces generalizes and gives a unified treatment of all these cases. There are primarily five books devoted to nonlinear potential theory, viz. Adams– Hedberg [5], Heinonen–Kilpeläinen–Martio [171], Malý–Ziemer [258], Mizuta [287] and Turesson [342]. They all study potential theory on unweighted ([5], [258] and [287]) or weighted ([171] and [342]) Rn . In [5] and [342] the focus is on higher order potential theory, whereas the main topic in [171] and [258] is the potential theory of p-harmonic functions (on weighted and unweighted Rn , respectively). The main focus in [287] is on Riesz potentials. The topics covered in our book are closest to [171], but there are also some parts in common with [258]. There is also some overlap, especially in Chapter 8, with the book by Giusti [146] (for unweighted Rn ). Let us also mention the survey papers by Martio [266], which has much in common with this book, and by Björn–Björn [47], which is based on an early version of this book. Moreover, the forthcoming monograph Heinonen–Koskela–Shanmugalingam– Tyson [177] has a certain overlap with the first part of this book. This book consists of two related parts. In the first part we develop the theory of Newtonian (Sobolev) spaces on metric spaces, and in the second part we develop the potential theory associated with p-harmonic functions on metric spaces. BoththeNewtonianandthe p-harmonictheoriesonmetricspaceshavenowreached such a maturity that we think they deserve to be written in book form. So far, both theories are scattered over a large number of different research papers published during
- 10. vi Preface the last two decades, with obvious difficulties for those interested in them. In fact, very few of the results in this book are available in book form. When writing a book, one is faced with many decisions on what to include. Natu- rally, the choice is influenced by the taste and interests of the authors. Throughout the book we consider solely real-valued Newtonian spaces. We also restrict ourselves to the theory of p-harmonic functions (defined through upper gradi- ents) on complete doubling metric spaces supporting a Poincaré inequality. We do not cover quasiminimizers, nor the noncomplete theory. Neither do we include results that only hold for Cheeger p-harmonic functions, or only in Rn . Related topics that could have been covered but were left out include: Differentia- bility in metric spaces (in particular, we do not prove Cheeger’s theorem (Theorem B.6)) and the Poincaré inequalities in the various examples discussed in Section 1.7 and Ap- pendix A. Including these topics would have made the book substantially different, and many of these topics would rather deserve books on their own. There are also many recent results which we have not been able to cover, nor have we included a proof of Keith–Zhong’s theorem, Theorem 4.30. (See however the appendices, notes and remarks for some references and comments on the topics mentioned above.) This book is reasonably self-contained and we develop both the Newtonian theory and the p-harmonic theory from scratch. Naturally, we cannot develop all the math- ematics needed for this book, and we have most often chosen to omit results which are available elsewhere in book form, but sometimes providing a reference. Thus, the reader is assumed to know measure theory and functional analysis. Apart from comparison results between Newtonian spaces and ordinary Sobolev spaces, there is no background needed in Sobolev space theory. In Chapters 1 and 2 we start developing the theory of upper gradients and Newtonian functions. Here we have collected the theory which works well in general metric spaces, i.e. without any assumptions such as doubling or the validity of a Poincaré inequality. In Chapter 3 we introduce the doubling condition and study some of its conse- quences. The reader interested only in the Newtonian theory can safely skip Sec- tions 3.3–3.5. The John–Nirenberg lemma and its consequences in Sections 3.3 and 3.4, will be used only in the proof of the weak Harnack inequality for superminimizers (The- orem 8.10). (More specifically, it is Corollary 3.21 which is used there.) The Gehring lemma in Section 3.5, although important, is not used in this book. In Chapter 4 we introduce Poincaré inequalities of various types and look at their relations and some consequences. We discuss, in particular, connections with quasi- convexity. In Chapters 5 and 6 we study various properties of Newtonian spaces, which follow from assuming doubling and a p-Poincaré inequality. Throughout Chapters 1–6 we have taken extra care to see when the proofs are valid for p D 1 and with minimal assumptions, in particular when completeness is not needed. However, as our main interest is in the case p 1, we have not dwelled further on the case p D 1 when it requires special proofs.
- 11. Preface vii The second part of the book consists of Chapters 7–14. In it we develop the potential theory associated with p-harmonic functions. Already from the beginning we need to use doubling, a p-Poincaré inequality and that p 1. We also assume completeness, and these assumptions are general throughout the second part. We have taken special care to cover the case when X is a bounded metric space. In Chapters 7 and 8 we study the basic properties of superminimizers and show interior regularity, whereas Chapter 9 is devoted to superharmonic functions. In Chapter 10 we look closely at the Dirichlet problem, and in Chapter 11 on bound- ary regularity. In Chapter 12 we consider removable singularities, which are in turn needed to obtain the trichotomy (Theorem 13.2) motivating the study and classification of irregular boundary points in Chapter 13. Finally, in Chapter 14 we show that open sets can be approximated by regular sets and give some consequences thereof, including another formulation of the Dirichlet problem. In Appendix A we give various examples of metric spaces satisfying our basic assumptions (completeness, doubling and the validity of a p-Poincaré inequality). In particular, we show that on weighted Rn our theory coincides with the one developed in Heinonen–Kilpeläinen–Martio [171] and other sources. When specialized to weighted Rn , many of the results in this book appear in [171]. We have refrained from pointing out exactly which ones in the notes at the end of each chapter. Instead, we make [171] as a general reference for comparison throughout the book. The reader is also referred to the comments and references in [171]. In any case, we would like to point out that many of the results in Chapters 10–14 do not appear in [171] (when specialized to weighted Rn ). In Appendix B we take a quick look at Hajłasz–Sobolev and Cheeger–Sobolev spaces, and in Appendix C we give a short overview of the more general potential theory of quasiminimizers. The reader interested in open problems should observe that the item open problem in the index gives references to all open problems stated in the book. We started writing this book when giving a graduate course on this topic during the autumn of 2005 in Linköping. In fact, we gave a similar course in Prague in the autumn of 2003, and the handwritten notes we then obtained are in a sense the very first draft of this book. We thank the participants for their active role in the courses: Jan Kališ, Jan Malý, Petr Přecechtěl and Jiří Spurný in Prague, and Gunnar Aronsson, Thomas Bäckdahl, Daniel Carlsson, David Färm, Thomas Karlsson, Mats Neymark, Björn Textorius, Johan Thim and Bengt-Ove Turesson in Linköping. Later on we have also given lecture series on these topics: Anders at the Paseky springschool FunctionSpaces, InequalitiesandInterpolation, inJune2009, onNewto- nian spaces, and Jana within the Taft Research Seminar at the University of Cincinnati, during the winter term 2010. Again, we thank all the participants. We are also grateful to Zohra Farnana, Heikki Hakkarainen, Stanislav Hencl, David Herron, Tero Kilpeläinen, Juha Kinnunen, Riikka Korte, Visa Latvala, Tero
- 12. viii Preface Mäkäläinen, Lukáš Malý, Niko Marola, Olli Martio, David Minda, Mikko Parviainen, Nageswari Shanmugalingam, Tomas Sjödin, Heli Tuominen and two anonymous ref- erees for many useful comments given on draft versions of this book, and to Irene Zimmermann for her patience with our typesetting requests. Particular thanks go to Outi Elina Kansanen and Juha Kinnunen who wrote draft versions for Sections 3.3–3.5. Last but not least, we are grateful to Juha Heinonen, who introduced us to analysis on metric spaces during his course inAnnArbor in 1999. We also encourage all future comments, in particular pointing out any mistakes, for which we apologize. Acknowledgement. The authors have been supported by various sources during their work on this book: the Swedish Science Research Council, Magnuson’s fund of the Royal Swedish Academy of Sciences, the Charles Phelps Taft Research Center at the University of Cincinnati, the Swedish Fulbright Commission, and Linköpings univer- sitet. They also belong to the European Science Foundation Networking Programme Harmonic and Complex Analysis and Applications and to the Scandinavian Research Network Analysis and Application. Linköping, September 2011 Anders and Jana Björn
- 13. Contents Preface v 1 Newtonian spaces 1 1.1 The metric space X and some notation . . . . . . . . . . . . . . . . 1 1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Upper gradients and the Newtonian space N 1;p . . . . . . . . . . . . 8 1.4 The Sobolev capacity Cp . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 p-weak upper gradients and modulus of curve families . . . . . . . . 15 1.6 Banach space and ACCp . . . . . . . . . . . . . . . . . . . . . . . . 24 1.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2 Minimal p-weak upper gradients 37 2.1 Fuglede’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2 Minimal p-weak upper gradients . . . . . . . . . . . . . . . . . . . 40 2.3 Calculus for p-weak upper gradients . . . . . . . . . . . . . . . . . 45 2.4 The glueing lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.5 N 1;p ./ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.6 N 1;p loc and Dp loc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.7 N 1;p 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.8 Gp loc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.9 Dependence on p in gu . . . . . . . . . . . . . . . . . . . . . . . . 59 2.10 Representation formulas for gu . . . . . . . . . . . . . . . . . . . . 61 2.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3 Doubling measures 65 3.1 Doubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2 The maximal function . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.3 BMO and John–Nirenberg’s lemma . . . . . . . . . . . . . . . . . . 70 3.4 Consequences of John–Nirenberg’s lemma . . . . . . . . . . . . . . 75 3.5 Gehring’s lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4 Poincaré inequalities 84 4.1 Poincaré inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.2 Characterizations of Poincaré inequalities . . . . . . . . . . . . . . . 88 4.3 BiLipschitz invariance . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.4 .q; p/-Poincaré inequalities . . . . . . . . . . . . . . . . . . . . . . 91 4.5 Quasiconvexity and connectivity . . . . . . . . . . . . . . . . . . . . 99
- 14. x Contents 4.6 Poincaré inequalities in quasiconvex spaces . . . . . . . . . . . . . . 103 4.7 Inner metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.8 The relation between L and . . . . . . . . . . . . . . . . . . . . . 110 4.9 Measurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5 Properties of Newtonian functions 116 5.1 Density of Lipschitz functions . . . . . . . . . . . . . . . . . . . . . 116 5.2 Quasicontinuity of Newtonian functions . . . . . . . . . . . . . . . . 123 5.3 Continuity of Newtonian functions . . . . . . . . . . . . . . . . . . 134 5.4 Density of Lipschitz functions in N 1;p 0 . . . . . . . . . . . . . . . . 138 5.5 Sobolev embeddings and inequalities . . . . . . . . . . . . . . . . . 141 5.6 Lebesgue points for N1;p -functions . . . . . . . . . . . . . . . . . . 147 5.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6 Capacities 154 6.1 Mazur’s lemma and its consequences . . . . . . . . . . . . . . . . . 154 6.2 Properties of Cp in complete doubling p-Poincaré spaces . . . . . . 157 6.3 The variational capacity capp . . . . . . . . . . . . . . . . . . . . . 161 6.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 7 Superminimizers 170 7.1 Introduction to potential theory . . . . . . . . . . . . . . . . . . . . 170 7.2 The obstacle problem . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.3 Definition of (super)minimizers . . . . . . . . . . . . . . . . . . . . 178 7.4 Convergence results for superminimizers . . . . . . . . . . . . . . . 183 7.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 8 Interior regularity 191 8.1 Weak Harnack inequalities for subminimizers . . . . . . . . . . . . . 191 8.2 Weak Harnack inequalities for superminimizers . . . . . . . . . . . . 197 8.3 Hölder continuity for p-harmonic functions . . . . . . . . . . . . . . 201 8.4 The need for in Harnack’s inequality . . . . . . . . . . . . . . . . 205 8.5 Lsc-regularized superminimizers . . . . . . . . . . . . . . . . . . . 206 8.6 Lsc-regularized solutions of obstacle problems . . . . . . . . . . . . 209 8.7 p-harmonic extensions . . . . . . . . . . . . . . . . . . . . . . . . . 212 8.8 A sharp weak Harnack inequality for superminimizers . . . . . . . . 213 8.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 9 Superharmonic functions 218 9.1 Definition of superharmonic functions . . . . . . . . . . . . . . . . . 218 9.2 Weak Harnack inequalities for superharmonic functions . . . . . . . 220 9.3 Lsc-regularity and the minimum principle . . . . . . . . . . . . . . . 222
- 15. Contents xi 9.4 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 9.5 Convergence results for superharmonic functions . . . . . . . . . . . 229 9.6 Harnack’s convergence theorem for p-harmonic functions . . . . . . 233 9.7 Comparison of sub- and superharmonic functions . . . . . . . . . . . 234 9.8 New superharmonic functions from old . . . . . . . . . . . . . . . . 235 9.9 Integrability of superharmonic functions . . . . . . . . . . . . . . . 238 9.10 Lebesgue points for superharmonic functions . . . . . . . . . . . . . 244 9.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 10 The Dirichlet problem for p-harmonic functions 249 10.1 Continuous boundary values . . . . . . . . . . . . . . . . . . . . . . 250 10.2 The Kellogg property . . . . . . . . . . . . . . . . . . . . . . . . . . 251 10.3 Perron solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 10.4 Resolutivity of Newtonian functions . . . . . . . . . . . . . . . . . . 255 10.5 Resolutivity of continuous functions . . . . . . . . . . . . . . . . . . 261 10.6 Some consequences of resolutivity . . . . . . . . . . . . . . . . . . . 263 10.7 Resolutivity of semicontinuous functions . . . . . . . . . . . . . . . 265 10.8 The p-harmonic measure . . . . . . . . . . . . . . . . . . . . . . . . 268 10.9 Poisson modification . . . . . . . . . . . . . . . . . . . . . . . . . . 271 10.10 The resolutivity problem . . . . . . . . . . . . . . . . . . . . . . . . 273 10.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 11 Boundary regularity 277 11.1 Barrier characterization of regular points . . . . . . . . . . . . . . . 277 11.2 Boundary regularity for the obstacle problem . . . . . . . . . . . . . 280 11.3 Characterizations of regularity . . . . . . . . . . . . . . . . . . . . . 285 11.4 The Wiener criterion . . . . . . . . . . . . . . . . . . . . . . . . . . 288 11.5 Regularity componentwise . . . . . . . . . . . . . . . . . . . . . . . 295 11.6 Fine continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 11.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 12 Removable singularities 303 12.1 Removability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 12.2 Nonremovability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 12.3 Removable sets with positive capacity . . . . . . . . . . . . . . . . . 312 12.4 Nonunique removability . . . . . . . . . . . . . . . . . . . . . . . . 314 12.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 13 Irregular boundary points 318 13.1 Semiregular and strongly irregular points . . . . . . . . . . . . . . . 318 13.2 Characterizations of semiregular points . . . . . . . . . . . . . . . . 320 13.3 Characterizations of strongly irregular points . . . . . . . . . . . . . 325 13.4 The sets of semiregular and of strongly irregular points . . . . . . . . 327
- 16. xii Contents 13.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 14 Regular sets and applications thereof 329 14.1 Regular sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 14.2 Wiener solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 14.3 Classically superharmonic functions . . . . . . . . . . . . . . . . . . 333 14.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 Appendices A Examples 337 A.1 N 1;p in Euclidean spaces . . . . . . . . . . . . . . . . . . . . . . . 337 A.2 Weighted Sobolev spaces on Rn . . . . . . . . . . . . . . . . . . . . 340 A.3 Uniform domains and power weights . . . . . . . . . . . . . . . . . 348 A.4 Glueing spaces together . . . . . . . . . . . . . . . . . . . . . . . . 349 A.5 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 A.6 Carnot–Carathéodory spaces and Heisenberg groups . . . . . . . . . 354 A.7 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357 A.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 B Hajłasz–Sobolev and Cheeger–Sobolev spaces 360 B.1 Hajłasz–Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . 360 B.2 Cheeger–Sobolev spaces and differentiable structures . . . . . . . . . 363 C Quasiminimizers 365 Bibliography 369 Index 389
- 17. Chapter 1 Newtonian spaces Throughout the book we will study potential theory on the metric space X. We make some general assumptions on X in the beginning of Section 1.1 below. These as- sumptions will hold throughout the book. In many of the chapters we will make some additional assumptions holding throughout that chapter. These assumptions will al- ways be listed at the very beginning of the chapter. Occasionally we will state some additional assumptions at the very beginning of a section. In the first half of this book, Chapters 1–6, we introduce and study Sobolev type spaces on metric measure spaces. These spaces, called Newtonian spaces, will be the natural setting for p-harmonic functions and potential theory in the second half of the book. In Appendix B we mention some other possible definitions of Sobolev type spaces on metric spaces and relate them to our definition. Let us first take a very quick look at first-order Sobolev spaces on Rn . For 1 p 1 and f 2 Lp .Rn / we define kf kp W 1;p.Rn/ D Z Rn .jf jp C jrf jp / dx; where rf is the weak or distributional gradient of f . The Sobolev space W 1;p .Rn / is then given by W 1;p .Rn / D ff W kf kW 1;p.Rn/ 1g: To define W 1;p .Rn /, one uses the gradient, i.e. the directional derivatives. In metric spaces we cannot talk about directions, so that is problematic. However, we do not really use the vector rf , only the scalar jrf j is necessary to define W 1;p .Rn /, and for jrf j there is a possible counterpart in metric spaces, called upper gradient and introduced by Heinonen–Koskela [173], [174]. Before defining upper gradients, in Section 1.3, we introduce our metric spaces and give some background results. Let us mention here that in Section 1.7 we give a short overview of some examples of metric spaces on which the theory of Newtonian spaces is useful. A more detailed discussion is given in Appendix A. 1.1 The metric space X and some notation We assume throughout the book that 1 p 1 and that X D .X; d; / is a metric space endowed with a metric d and a positive complete Borel measure such that .B/ 1 for all balls B X (we make the convention that balls are nonempty and open).
- 18. 2 1 Newtonian spaces We emphasize that to start with we do not assume that X is complete nor locally compact. We will however assume that X contains at least two points (to avoid special formulations needed to cover the pathological cases when X is empty and when X is a singleton). We also emphasize that the -algebra on which is defined is obtained by the completion of the Borel -algebra. We further extend as an outer measure on X, so that for an arbitrary set A X we have .A/ D inff.E/ W E A is a Borel setg: Proposition 1.1. The measure is Borel regular, i.e. for every A X there is a Borel set E A such that .A/ D .E/. Note that this is the definition of Borel regularity used by Federer [123], Sec- tion 2.2.3, Mattila [275], Definition 1.5, and Heinonen [169], p. 3, whereas Rudin [311] has a more restrictive definition of Borel regularity which is not always fulfilled for our spaces. Proof. If .A/ D 1, then we may choose E D X. Otherwise we can find Borel sets Ej A with .Ej / .A/ C 1=j , j D 1; 2; ::: . Letting E D T1 jD1 Ej completes the proof. Let us here record some notation used throughout the book. We use the following notation for balls B.x0; r/ WD fx 2 X W d.x; x0/ rg; and for B D B.x0; r/ and 0, we let B D B.x0; r/. Further, we always assume that , 0 , 00 and t , t 2 R, are nonempty open subsets of X. In the metric space X, a.e. always means almost everywhere with respect to the given measure , unless otherwise stated. For a measurable set A X with 0 .A/ 1, we define the mean-value integral (or integral average) of f over A as « A f d WD 1 .A/ Z A f d; whenever the right-hand side exists, which it does, in particular, if f 2 L1 .A/ or if f is nonnegative and measurable on A. We shall also use the abbreviation fA WD « A f d: In general we allow semicontinuous functions to be extended real-valued, i.e. to take values in x R D Œ1; 1. On the other hand, continuous functions are always assumed to be real-valued, unless mentioned otherwise.
- 19. 1.2 Preliminaries 3 We denote the characteristic function of a set E by E , and let sup ¿ D 1 and inf ¿ D 1. Further, we let fC D maxff; 0g and f D maxff; 0g so that f D fC f and jf j D fC C f. We define 0 1 WD 0 .1/ WD 0. This will, in particular, be useful when multi- plying a function by a characteristic function so that we always let fE D 0 outside of E, even when f may take the values ˙1. Moreover, we make the convention that 1 1 D 1 and .1/ .1/ D 1, see the comments after Definition 1.13 for the usefulness of this. For sequences of numbers or sets we will use the words increasing and decreasing in their nonstrict sense, so that e.g. the sequence faj g1 jD1 is increasing if ajC1 aj , j D 1; 2; ::: . We also say that E b A if x E is a compact subset of A, and let Lipc.A/ D ff 2 Lip.A/ W supp f b Ag be the set of Lipschitz functions on A with compact support. We shall later see that this class of functions will play the important role of test functions for us. Throughout the book we present a number of examples with X being Rn , n 1, or some subset thereof. In these cases, unless otherwise stated, we always equip X with the Euclidean metric and the natural Lebesgue measure (or restrictions thereof). Finally, we make the convention that, unless otherwise stated, the letter C denotes various positive constants whose exact values are not important and may vary with each usage. 1.2 Preliminaries We will need the following consequence of the Borel regularity. Proposition 1.2. If f W X ! x R is measurable, then there exist Borel functions f1; f2 W X ! x R such that f1 f f2 and f1 D f2 a.e. A Borel function f W X ! Y is by definition a function such that f 1 .G/ X is a Borel set for every open set G Y . It follows that f 1 .E/ X is a Borel set for every Borel set E Y . The following simple lemma will be useful in the proof of Proposition 1.2. Lemma 1.3. Let f W X ! x R. Then f is a Borel function if and only if Eq D fx W f .x/ qg is a Borel set for every q 2 Q.
- 20. 4 1 Newtonian spaces Proof. The necessity is clear, so let us look at the sufficiency. Let I D .a; b/ R, a b, be an open interval. Then f 1 .I/ D [ Q3qa Eq n Q3qb Eq is a Borel set. Moreover, f 1 .1/ D q2Q Eq and f 1 .1/ D X n [ q2Q Eq are Borel sets. We conclude that f 1 .I/ is a Borel set for any open interval in x R (which is now allowed to contain the points ˙1). Let now E x R be open. Then E can be written as a countable union of pairwise disjoint open intervals, E D S1 jD1 Ij (where Ij is allowed to contain ˙1), see Lemma 1.4 below. Since f 1 .E/ D S1 jD1 f 1 .Ij / is a Borel set, we conclude that f is a Borel function. Proof of Proposition 1.2. For q 2 Q let Eq D fx W f .x/ qg. Let further A0 q be a Borel set such that A0 q Eq and .A0 q nEq/ D 0, and let Aq D T Q3q0q A0 q0 which is a Borel set. As A0 q0 Eq0 Eq for every q0 q, q0 2 Q, we see that Eq Aq A0 q and thus .Aq n Eq/ .A0 q n Eq/ D 0. Note that Aq Aq0 if q0 q, q0 2 Q. Let now f2.x/ D supfq 2 Q W x 2 Aqg supfq 2 Q W x 2 Eqg D f .x/: Then f2 is a Borel function by Lemma 1.3. Moreover, fx W f2.x/ ¤ f .x/g D fx W f2.x/ f .x/g D [ q2Q fx W f2.x/ q f .x/g D [ q2Q fx W x 2 Aq and x … Eqg D [ q2Q .Aq n Eq/; which is a countable union of sets of measure zero and hence itself of measure zero. Thus f2 has the required properties. The construction of f1 is similar, but its existence may also be obtained by applying the already proved half of the proposition to the function f . Lemma 1.4. Let G x R be open. Then G can be written as a countable (or finite) union of pairwise disjoint open intervals (which are allowed to contain ˙1). This lemma is well known, but as the proof is short and we have not found a good reference we give the proof here. Let us point out, for later use, that a countable set may be finite.
- 21. 1.2 Preliminaries 5 Proof. Write G as a union of its components. Each component is necessarily an open interval. Associate with each component one of its rational points. Then we see that the set of components cannot have larger cardinality than Q. Proposition 1.5. Let Y be a metric space. Then the following are equivalent: (a) Y is separable; (b) Y is second countable, i.e. it has a countable open base; (c) Y is Lindelöf, i.e. every open cover of Y has a countable subcover. It is easily seen that these properties are also inherited by any subset of Y (in the induced topology). This is well known, see e.g. Exercise 5, p. 194, in Munkres [296], or Figure 10 in Steen–Seebach [327]. However, we have not found one good reference with full proofs of all implications. Proof. (a) , (b) See Theorem 2, p. 177, in Kuratowski [235]. (b) ) (c) See Theorem 2, p. 176, in [235]. (c) ) (b) See the second paragraph on p. 35 in Steen–Seebach [327]. In our considerations on the metric measure space X, the support of will be of main interest. The following result is therefore useful. See also Proposition 1.53 and the comment before it. Note, however, that we do not want to assume that X D supp in general, since we want to be able to consider arbitrary subsets of X as metric spaces on their own and it then may not be true that the support of the restriction of the underlying measure is the whole subset under consideration. Recall that x 2 supp , the support of , if and only if .B.x; r// 0 for every r 0. Or in other terms X n supp is the union of all balls with zero measure, and is the largest open set with zero measure, in the sense that it contains any open set with zero measure. (The support of a function is similarly defined as the complement of the largest open set on which f is 0.) Proposition 1.6. The metric space supp is separable, second countable and Lindelöf. Before proving Proposition 1.6, let us make some remarks. The following simple but very useful covering lemma is well known. Lemma 1.7 (5-covering lemma). Let B be a family of balls in X with uniformly bounded radii. Then there exists a subfamily B0 B of pairwise disjoint balls such that [ B2B B [ B2B0 5B: If X is separable, then the subfamily B0 can be chosen to be countable.
- 22. 6 1 Newtonian spaces Proof. For a proof of the first part see e.g. Ambrosio–Tilli [17], Theorem 2.2.3, Hei- nonen [169], Theorem 1.2, or Federer [123], Theorem 2.8.4 (choose D 2 and let ı be the radius of the balls in Federer’s formulation). For the second part assume that X is separable. Then X is Lindelöf by Proposi- tion 1.5, and hence we can find a countable subfamily B00 B0 such that S B2B00 5B D S B2B0 5B. Remark 1.8. The constant 5 in the 5-covering lemma is not optimal, but has become standard as one wants to have as simple an argument as possible. With just a slight extra thought most proofs actually show that one may replace 5 by 3 C for any 0. In Federer [123], Theorem 2.8.4, this is even a consequence of the statement, just choose D 1 C 1 2 . The following example shows that one cannot replace 5 by 3. Example 1.9. Let X D R and B2j1 D B.22=j; 21=j/ for j D 1; 2; ::: . In other terms, B2j1 is the interval .1=j; 4 3=j/. Let also B2j D B2j1, j D 1; 2; ::: . Then 0 2 Bj for all j D 1; 2; ::: , and thus any disjoint subfamily must consist of just one ball, say Bk. If k D 2j 1 is odd, then 3Bk is the interval .4 C 1=j; 8 5=j/ and therefore does not cover S1 jD1 Bj D .4; 4/. The case when k is even is similar, and hence the constant 5 in the 5-covering lemma cannot be replaced by 3. Proof of Proposition 1.6. Fix x0 2 Y WD supp and let 0 and R 0. Let B D fB.x; / W x 2 Y B.x0; R/g: By the 5-covering lemma (Lemma 1.7), we can find a subfamily B0 B of pairwise disjoint balls so that Y B.x0; R/ [ B2B B [ B2B0 5B: As X B2B0 .B/ D [ B2B0 B .B.x0; R C // 1; and .B/ 0 for every B 2 B0 , we conclude that B0 is a countable set. The centres of the balls B 2 B0 make a countable -net in Y B.x0; R/. Letting D 1=n, n D 1; 2; ::: , shows that Y B.x0; R/ is separable. Thus Y D S1 jD1.Y B.x0; j// is also separable. That Y is second countable and Lindelöf follows from Proposition 1.5. Lemma 1.10 (Cavalieri’s principle). Let be a measure on X. If f W X ! Œ0; 1 is a -measurable function and 0 q 1, then Z X f q d D q Z 1 0 q1 .fx W f .x/ g/ d:
- 23. 1.2 Preliminaries 7 The Cavalieri principle is a standard tool in integration theory, we state it here for the reader’s convenience. Sometimes it is the case q D 1 that is called the Cavalieri principle, but the general case follows after the change of variable 7! q . The Cavalieri principle can be proved in several different ways, e.g. using the Fubini theorem. The most convenient way actually depends on how the Lebesgue integral is defined and the proof truly belongs to integration theory. We refer the reader in need to Folland [124], Proposition 6.24. Another result which eventually will be needed is that continuous functions can be approximated by Lipschitz functions on compact sets. Recall that f W Y ! R is Lipschitz on the metric space .Y; d/ if there exists a constant C such that jf .x/ f .y/j Cd.x; y/ for all x; y 2 Y: Proposition 1.11. Let .Y; d/ be a compact metric space and let u 2 C.Y / and 0. Then there is v 2 Lip.Y / such that supy2Y ju.y/ v.y/j . This follows from the Stone–Weierstrass theorem (see e.g. Rudin [312], p. 122), but for the reader’s convenience, we give an elementary proof of this fact. Proof. We can assume that minY u D 0 and let Gk D fx 2 Y W u.x/ kg for positive integers k. Clearly, only finitely many Gk are nonempty. As Y is compact, so is each x Gk, and hence there exists ı 0 so that dist.Gk; Y n Gk1/ ı whenever Gk ¤ ¿. Let, for x 2 Y , u.x/ D X k1 1 dist.Gk; x/ ı C ; where we only sum over nonempty Gk. Then u 2 Lip.Y / (with constant =ı) and ju.x/ u.x/j for all x 2 Y . We will also need to approximate lower semicontinuous functions by Lipschitz functions from below. Proposition 1.12. Let .Y; d/ be a compact metric space and let uW Y ! .1; 1 be lower semicontinuous. Then there exists an increasing sequence fuj g1 jD1 of Lipschitz functions such that uj .x/ ! u.x/, as j ! 1, for all x 2 Y . If u 0, then we may choose uj 0, j D 1; 2; :::. In fact it is fairly easy to see that the uj constructed below is the largest j -Lipschitz function u (apart from the case when u 1 when no such largest function exists). Proof. As u is lower semicontinuous on a compact set, it attains its minimum, which by assumption is not 1. Hence, without loss of generality, we may assume that u 0. If u 1, then we can take uj j, j D 1; 2; ::: , and we therefore exclude this case below. Fix a positive integer j, and let uj .x/ D inf y2Y .u.y/ C jd.x; y//:
- 24. 8 1 Newtonian spaces It is easy to see that uj is real-valued and that 0 uj ujC1 u. Let us first show that uj is j-Lipschitz. Let x; y 2 Y and 0. Then we can find z 2 Y such that uj .x/ u.z/ C jd.z; x/ . It follows that uj .y/ u.z/ C jd.z; y/ u.z/ C jd.z; x/ C jd.x; y/ uj .x/ C jd.x; y/ C : Letting ! 0 shows that uj .y/ uj .x/ C jd.x; y/. As we also have uj .x/ uj .y/ C jd.x; y/, we have shown that uj is j-Lipschitz. Let x 2 Y be arbitrary. It remains to show that uj .x/ ! u.x/, as j ! 1. Let u.x/. As u is lower semicontinuous we can find a ball B D B.x; r/ in Y such that u in B. For j =r we have uj .x/ D min inf y2B .u.y/ C jd.x; y//; inf y2Y nB .u.y/ C jd.x; y// min inf y2B ; inf y2Y nB jr D : Hence uj .x/ ! u.x/, as j ! 1. 1.3 Upper gradients and the Newtonian space N1;p In this section we introduce upper gradients as a substitute for the modulus of the usual gradient. We will later see that under some assumptions it is possible to control a function by its upper gradients. On the other hand, in situations as in Example 1.22 the upper gradients do not give any information about the function. To begin with, we will leave this question aside and just concentrate on developing the theory of upper gradients, and p-weak upper gradients, whatever use they may have. In Chapters 3 and 4 we will introduce some conditions that will be sufficient for the upper gradients to control their functions. By a curve in X we will mean a rectifiable nonconstant continuous mapping from a compact interval. (For us only such curves will be interesting, in general a curve is a continuous mapping from an interval. A rectifiable curve is a curve with finite length.) A curve can thus be parameterized by arc length ds, and we will always assume that all curves are parameterized by arc length, see e.g. Ambrosio–Tilli [17], Section 4.2, or Heinonen [169], Section 7.1. Note that every curve is Lipschitz continuous with respect to its arc length parameterization. We let .X/ denote the family of all curves on X. By abuse of notation we also denote the image of a curve by . Definition 1.13. A nonnegative Borel function g on X is an upper gradient of an extended real-valued function f on X if for all curves W Œ0; l ! X, jf . .0// f . .l //j Z g ds: (1.1)
- 25. 1.3 Upper gradients and the Newtonian space N 1;p 9 Recall our convention that 1 1 D 1 and .1/ .1/ D 1, so that R g ds D 1 if at least one of f . .0// and f . .l // is infinite. Note that for a Borel function g 0, the integral R g ds is defined (with a value in Œ0; 1) for all curves , since g ı W Œ0; l ! x R is a nonnegative Borel function. Note also that upper gradients are not unique. In particular, by adding a nonnegative Borel function to an upper gradient of f we obtain a new upper gradient of f . The function g 1 is an upper gradient of every function. The following simple proposition provides us with plenty nontrivial examples of upper gradients. Proposition 1.14. If f W X ! R is locally Lipschitz, then the lower pointwise dilation lip f .x/ D lim inf r!0 sup y2B.x;r/ jf .y/ f .x/j r (1.2) is an upper gradient of f . Proof. Let W Œ0; l ! X be a curve. Since is parameterized by arc length, it is 1-Lipschitz, and thus f ı is Lipschitz and hence absolutely continuous on Œ0; l (see Definition 1.57 for the definition of absolute continuity). We thus have jf . .0// f . .l //j Z l 0 j.f ı /0 .t/j dt Z l 0 .lip f / ı .t/ dt D Z lip f ds: Corollary 1.15. If X D Rn and f 2 C1 loc.Rn /, then jrf j is an upper gradient of f . Proof. For all x 2 Rn and all y 2 B.x; r/ we have jf .y/ f .x/j D jru.x/ .y x/ C o.jy xj/j r jru.x/j C o.r/: Inserting this into the definition (1.2) of lip f .x/ finishes the proof. This result will be extended to locally Lipschitz functions in Corollary 1.47, and further refined in Propositions A.3 and A.11. Note that if g and g0 are upper gradients of u and v, respectively, then g g0 is in general not an upper gradient of u v. (An easy example is to let u.x/ D x, v.x/ D x and g D g0 1 on R, in which case g g0 0 is not an upper gradient of .u v/.x/ D 2x.) However, we have the following subadditivity result. Lemma 1.16. Let g and g0 be upper gradients of u and v, respectively, and a 2 R. Then jajg and g C g0 are upper gradients of au and u C v, respectively. Proof. This follows immediately from Definition 1.13. Now that we have upper gradients, it is possible to define analogues of Sobolev spaces on metric measure spaces.
- 26. 10 1 Newtonian spaces Definition 1.17. Whenever u 2 Lp .X/, let kukN1;p.X/ D Z X jujp d C inf g Z X gp d 1=p ; where the infimum is taken over all upper gradients g of u. The Newtonian space on X is the space N 1;p .X/ D fu W kukN1;p.X/ 1g: Let us here point out that we assume that functions are defined everywhere, and not just up to an equivalence class in the corresponding function space. When we say that u 2 N 1;p .X/ we thus assume that u is a function defined everywhere. Remark 1.18. A minor point here which is easily overlooked is that one of course can use an equivalent norm, such as kukLp C infg kgkLp , when studying N 1;p .X/. However, when we define the capacity in Definition 1.24 it is important to use our norm to get a subadditive capacity. In some situations we will also have use for some related spaces. Definition 1.19. Let us define the following spaces z N1;p .X/ D N 1;p .X/= ; where u v if and only if ku vkN1;p.X/ D 0, and y N 1;p .X/ D fu W u D v a.e. for some v 2 N 1;p .X/g: Note that in ku vkN1;p.X/, the infimum is taken over upper gradients of u v, not over upper gradients of u and v. We equip z N 1;p .X/ and y N1;p .X/ with the norms induced by N 1;p .X/ (so that kuk y N 1;p.X/ D kvkN1;p.X/ if y N 1;p .X/ 3 u D v 2 N 1;p .X/ a.e.). It is not obvious that the norm on y N 1;p .X/ is well defined, but this follows from Lemma 1.62 below. Let us already here point out that the equivalence classes in z N 1;p .X/ are not up to measure zero, but are finer than that, in fact, as we will see, they are up to capacity zero. In Proposition 1.65 we will find that y N1;p .X/= ae Š z N 1;p .X/, where u ae v if u D v a.e. The space y N1;p .X/= ae is closer to traditional Sobolev spaces since it has a.e.-equivalence classes. It will however turn out to be both useful and important for us to have the more refined q.e.-equivalence classes in z N 1;p .X/ (i.e. to study the functions in N1;p .X/), see e.g. the discussion following Proposition 1.65. That N 1;p .X/, z N 1;p .X/ and y N 1;p .X/ are vector spaces follows directly from Lemma 1.16. The only difficulty with showing that k kN1;p.X/ is a norm on z N 1;p .X/ (and a seminorm on N 1;p .X/) is to prove the triangle inequality. To see this, let u; v 2
- 27. 1.3 Upper gradients and the Newtonian space N 1;p 11 N1;p .X/ and 0 be arbitrary. Find upper gradients g; g0 2 Lp .X/ of u and v, respectively, so that .kukp Lp.X/ C kgkp Lp.X/ /1=p kukN1;p.X/ C ; .kvkp Lp.X/ C kg0 kp Lp.X/ /1=p kvkN 1;p.X/ C : (1.3) By Lemma 1.16, g C g0 is an upper gradient of u C v. Now, note that the left-hand sides in (1.3) are lp -norms (on R2 ) of .kukLp.X/; kgkLp.X// and .kvkLp.X/; kg0 kLp.X//; respectively. Similarly, ku C vkN1;p.X/ .ku C vkp Lp.X/ C kg C g0 kp Lp.X/ /1=p ..kukLp.X/ C kvkLp.X//p C .kgkLp.X/ C kg0 kLp.X//p /1=p ; which is the lp -norm of .kukLp.X/ C kvkLp.X/; kgkLp.X/ C kg0 kLp.X//: The triangle inequality for the lp -norm now implies that ku C vkN1;p.X/ .kukp Lp.X/ C kgkp Lp.X/ /1=p C .kvkp Lp.X/ C kg0 kp Lp.X/ /1=p kukN 1;p.X/ C C kvkN 1;p.X/ C and letting ! 0 proves the triangle inequality for k kN1;p.X/. The completeness of z N 1;p .X/, i.e. that it is a Banach space, is harder to prove and will be obtained in Theorem 1.71. Theorem 1.20. The space N 1;p .X/ is a lattice, i.e. if u; v 2 N 1;p .X/, then maxfu; vg; minfu; vg 2 N 1;p .X/: Proof. Let w D maxfu; vg 2 Lp .X/. Let further g; g0 2 Lp .X/ be upper gradients of u and v, respectively, and g00 WD g C g0 2 Lp .X/. Then for a curve W Œ0; l ! X we have jw. .0// w. .l //j ju. .0// u. .l //j C jv. .0// v. .l //j Z g ds C Z g0 ds D Z g00 ds; and thus g00 is an upper gradient of w, and w 2 N 1;p .X/. The proof for minfu; vg is similar.
- 28. 12 1 Newtonian spaces With a little more care one easily sees that maxfg; g0 g is an upper gradient of minfu; vg and of maxfu; vg. In Corollary 2.20 we give more precise information about the best upper gradients of minfu; vg and maxfu; vg. The following is an immediate consequence of Theorem 1.20. Corollary 1.21. Assume that u 2 N 1;p .X/. Then uC; u; juj 2 N 1;p .X/. The following examples show that in some cases the theory of upper gradients becomes quite pathological. Example 1.22. If the space X contains no nonconstant rectifiable curves, e.g. if X is discrete (or more generally totally disconnected), or the von Koch snowflake curve (see Example 1.23 below), then g 0 is an upper gradient of any function. Hence it follows that N 1;p .X/ D Lp .X/. Example 1.23. In this example we consider the von Koch snowflake curve, which is a famous example of a curve of infinite length containing no rectifiable curves. Let K0 R2 , the 0th generation, be an equilateral triangle with side length 1. For each of the three sides, split it into three intervals of equal length and replace the middle one I by two sides I0 and I00 of an equilateral triangle (with sides I, I0 and I00 ) outside K0. We have thus produced the 1st generation K1 of the von Koch snowflake curve consisting of 12 pieces of length 1 3 each. Continuing in this way we obtain the nth generation Kn consisting of 3 4n pieces, each of length 3n . In the limit we obtain the von Koch snowflake curve K, which can formally be defined as K D fx 2 R2 W for every 0 there is n 1= and y 2 Kn so that jx yj g; or, in other terms, K is the Hausdorff limit of Kn, as n ! 1. We equip K with the distance from R2 . In this case, any curve (in the traditional sense) between two distinct points on K has infinite length and thus is not rectifiable. Hence there are no nonconstant rectifiable curves, and as in Example 1.22, g 0 is an upper gradient of any function, and N 1;p .K/ D Lp .K/. This in fact happens regardless of which measure we equip K with. For instance, one can equip K with the d-dimensional Hausdorff measure, with d D log 4=log 3, making it into an Ahlfors d-regular space, see Definition 3.4. 1.4 The Sobolev capacity Cp Definition 1.24. The capacity of a set E X is the number Cp.E/ D inf kukp N 1;p.X/ ; where the infimum is taken over all u 2 N 1;p .X/ such that u 1 on E.
- 29. 1.4 The Sobolev capacity Cp 13 This capacity is sometimes referred to as the Sobolev capacity. We say that a property regarding points in X holds quasieverywhere (q.e.) if the set of points for which it fails has capacity zero. Just as sets of zero measure are important in integration theory, sets of zero capacity will be important to us. Most of the time it is the distinction between sets of zero and positive capacity that will matter, though there are exceptions when the actual value of the capacity is important, e.g. in the Wiener criterion and the fine topology, see Sections 11.4 and 11.6. For most parts of the book we have p fixed and do not discuss dependence on p. However, the dependence of the capacity on p is discussed a little in Section 2.9. Remark 1.25. Maybe some words on our choice of terminology can be useful. The capacity depends on p, but we have refrained from making this dependence explicit in the notation and thus do not write “p-capacity” nor “p-q.e.”. This is in contrast to “p-modulus”, “p-a.e. curve” and “p-weak upper gradient” for which we have decided to make the p explicit. This is partly because they are less used and the dependence on p might otherwise be overlooked by some readers; with “capacity” and “q.e.” we see no such risk. In the later part of the book, Chapters 7–14, we have again refrained from making the dependence on p explicit when writing “minimizer”, “subminimizer”, “supermin- imizer”, “subharmonic” and “superharmonic”. We however do write “p-harmonic” as “harmonic” sounds too linear to us. When calculating capacities it is often convenient to use the following result. Some- times we will use the intermediate consequence that the capacity is obtained by taking the infimum over functions u E . Under additional assumptions on X, it is possible to further restrict the collection of functions which are used to test the capacity, see Theorem 6.7. Proposition 1.26. Let E X. Then Cp.E/ D inf kukp N 1;p.X/ ; where the infimum is taken over all u 2 N 1;p .X/ such that E u 1 on X. Proof. This follows easily by truncation. Let u 2 N 1;p .X/ be such that u 1 on E. Let further v D minfu; 1gC, W Œ0; l ! X be a curve, and g be an upper gradient of u. Then jv. .0// v. .l //j ju. .0// u. .l //j Z g ds; and thus g is an upper gradient also of v. Since jvj juj we have that kvkN1;p.X/
- 30. 14 1 Newtonian spaces kukN1;p.X/. It thus follows that Cp.E/ inf E v1 kvkp N1;p.X/ inf vDminfu;1gC u1 on E kvkp N 1;p.X/ inf u1 on E kukp N 1;p.X/ D Cp.E/: The capacity satisfies a number of properties. To begin with, we will need the following properties. Theorem 1.27. Let E; E1; E2; ::: be arbitrary subsets of X. Then (i) Cp.¿/ D 0; (ii) .E/ Cp.E/; (iii) if E1 E2, then Cp.E1/ Cp.E2/; (iv) Cp is countably subadditive and is also an outer measure, i.e. Cp 1 [ iD1 Ei 1 X iD1 Cp.Ei /: In order to prove (iv) we need the following lemma. Lemma 1.28. Let ui 1, i D 1; 2; :::, be functions with upper gradients gi , and let u D supi ui and g D supi gi . Then g is an upper gradient of u. Remark 1.29. Observe that if we remove the assumption that the functions ui be uniformly bounded from above, then the lemma becomes false. This is due to the special treatment of the function values ˙1 in Definition 1.13, and is most easily seen by letting ui i, u 1, g gi 0, and observing that g is not an upper gradient of u (if there are nonconstant curves in X). See also Lemma 1.52. Proof. Let W Œ0; l ! X be a curve. Let us first observe that u. .0// u. .l // D sup i .ui . .0// u. .l /// sup i .ui . .0// ui . .l ///: Together with the corresponding inequality for u. .l // u. .0// we get that ju. .0// u. .l //j sup i jui . .0// ui . .l //j sup i Z gi ds Z g ds: Proof of Theorem 1.27. The proofs of (i)–(iii) are trivial. (iv) We may assume that the right-hand side is finite. Let 0. Choose ui with Ei ui 1 and upper gradients gi such that kui kp Lp.X/ C kgi kp Lp.X/ Cp.Ei / C 2i :
- 31. 1.5 p-weak upper gradients and modulus of curve families 15 Let u D supi ui and g D supi gi . By Lemma 1.28, g is an upper gradient of u. Clearly u 1 on S1 iD1 Ei . Hence Cp 1 [ iD1 Ei kukp N 1;p.X/ Z X sup i ui p d C Z X sup i gi p d Z X 1 X iD1 up i d C Z X 1 X iD1 gp i d D 1 X iD1 Z X up i d C Z X gp i d 1 X iD1 Cp.Ei / C 2i D C 1 X iD1 Cp.Ei /: Letting ! 0 completes the proof of (iv). An easy, but interesting, consequence of the definition of the capacity is the follow- ing result. Proposition 1.30. If u 2 N 1;p .X/, then Cp.fx W ju.x/j D 1g/ D 0. Proof. Let E˙ D fx W u.x/ D ˙1g. Then u=k 1 on EC for all k 0. Thus Cp.EC/ u k p N1;p.X/ D kukp N1;p.X/ kp ! 0; as k ! 1: Similarly Cp.E/ D 0 (consider the function v WD u). Thus, by Theorem 1.27 (iv), Cp.fx W ju.x/j D 1g/ Cp.EC/ C Cp.E/ D 0. 1.5 p-weak upper gradients and modulus of curve families The following example illustrates a drawback of upper gradients, viz. that they are not preserved by Lp -convergence. This will make it necessary for us to study p-weak upper gradients. Example 1.31. Let E D f0g X D R2 and 1 p 2. Let also f D E and gj .x/ D ´ 1=j jxj; jxj 1; 0; jxj 1; j D 1; 2; ::: :
- 32. 16 1 Newtonian spaces Then gj is an upper gradient of f and since kgj kLp.X/ ! 0, as j ! 1, we see that kf kN1;p.X/ D 0 (and also that Cp.E/ D 0). However, the zero function is not an upper gradient of f , nor is, in fact, any function g such that g D 0 a.e. To see this, observe that if g D 0 a.e., then for a.e. ˛, g D 0 a.e. on ˛, where ˛.t/ D .t cos ˛; t sin ˛/, 0 t 1, by Fubini’s theorem (in polar coordinates). For such ˛ we have jf . .0// f . .1//j D 1 0 D Z ˛ g ds; showing that g is not an upper gradient of f . Thus the set of upper gradients of f is not a closed subset of Lp .X/ (or more correctly of the cone Lp C.X/ of nonnegative functions in Lp .X/). To overcome this complication we introduce p-weak upper gradients below. Later, in Section 2.2, we will show that any Newtonian function has a minimal p-weak upper gradient. (In the example above one can alternatively use the upper gradients Q gj .x/ D ´ 1=jxj; jxj 1=j; 0; jxj 1=j; j D 1; 2; ::: :/ Definition 1.32. A nonnegative measurable function g on X is a p-weak upper gradient of an extended real-valued function f on X if for p-a.e. curve W Œ0; l ! X (i.e. with the exception of a curve family of zero p-modulus, see below), jf . .0// f . .l //j Z g ds: (1.4) It is implicitly assumed that g is such that R g ds is defined (with a value in Œ0; 1) for p-a.e. curve. However, this is actually a consequence of the fact that g is assumed to be nonnegative and measurable, as we will see in Lemma 1.43. Recall the convention that 1 1 D 1, see also the comments after Defini- tion 1.13. Definition 1.33. Let be a family of curves on X. Then we define the p-modulus of by Modp. / D inf Z X p d; where the infimum is taken over all nonnegative Borel functions such that R ds 1 for all 2 . By Proposition 1.2, we may as well take the infimum over all nonnegative measur- able functions such that R ds 1 for all 2 . (In fact, using Proposition 1.37 it is easy to show that the infimum can as well be taken over all nonnegative measurable
- 33. 1.5 p-weak upper gradients and modulus of curve families 17 functions such that R ds 1 for p-a.e. 2 , but that cannot be used as the definition as the reasoning would be circular.) Just as with the capacity, it is whether the p-modulus is zero or not that will be important to us. The actual value plays a role only in (b) below, but this is a result we will not need; for us the weaker Corollary 1.38 will be sufficient. The reader who so wishes may therefore skip (b) below and its proof. Lemma 1.34. The following are true: (a) if 1 2, then Modp. 1/ Modp. 2/; (b) Modp S1 j D1 j P1 jD1 Modp. j /; (c) if for every 2 there exists a subcurve 0 2 0 of , then Modp. / Modp. 0 /. For the last part we need to make the following definition. Definition 1.35. A curve 0 is a subcurve of a curve W Œ0; l ! X if, after reparam- eterization and possibly reversion, 0 is equal to jŒa;b for some 0 a b l . We also say that g is an upper gradient of f along if jf . 0 .0// f . 0 .l0 //j Z 0 g ds for every subcurve 0 W Œ0; l0 ! X of . Note that, at a first glance, Lemma 1.34 (c) may look counterintuitive. However, as the proof shows, it is actually very natural. A simple illustration of the situation is provided by the following example. Example 1.36. Let X D R, D f g and 0 D f 0 g, where W Œ0; 2 ! R, 0 W Œ0; 1 ! R and .t/ D 0 .t/ D t. Then 0 is a subcurve of and and 0 satisfy the assumptions of Lemma 1.34 (c). For every admissible in the definition of Modp. /, we have 1 Z 2 0 dt 211=p Z 2 0 p dt 1=p ; with equalities for 1 2 . It follows that Modp. / D 21p 1 D Modp. 0 /; where Modp. 0 / is calculated similarly. This also shows that, roughly speaking, the longer the curves in , the smaller Modp. /.
- 34. 18 1 Newtonian spaces Proof of Lemma 1.34. (a)Thisistrivial, sincetheinfimuminthedefinitionofModp. 1/ is taken over a larger set than the infimum defining Modp. 2/. (b) Let 0 and let j be a nonnegative Borel function such that Z j ds 1 for all 2 j and such that R X p j d Modp. j / C =2j . Let D supj j . Then Modp 1 [ jD1 j Z X p d D Z X sup j j p d Z X 1 X jD1 p j d D 1 X jD1 Z X p j d 1 X jD1 Modp. j / C : Letting ! 0 completes the proof of this part. (c) Let 0 and let be a nonnegative Borel function such that Z ds 1 for all 2 0 and such that R X p d Modp. 0 / C . Let next 2 . Then there is a subcurve 0 2 0 of . Hence Z ds Z 0 ds 1: Thus Modp. / R X p d Modp. 0 / C ! Modp. 0 /, as ! 0. Proposition 1.37. Let x 2 X. The following are equivalent: (a) Modp. / D 0; (b) there is a nonnegative Borel function 2 Lp .X/ such that R ds D 1 for all 2 ; (c) there is a nonnegative such that 2 Lp .B.x; j// for all j D 1; 2; :::, and such that R ds D 1 for all 2 . See also Proposition 2.33 for another equivalent condition. Proof. (a) ) (b) For n D 1; 2; ::: there is a nonnegative Borel function n such that k nkLp.X/ 2n and Z n ds 1 for all 2 : Let D P1 nD1 n 2 Lp .X/. Then Z ds D 1 for all 2 :
- 35. 1.5 p-weak upper gradients and modulus of curve families 19 (b) ) (c) This is trivial. (c) ) (a) By Proposition 1.2 there is a nonnegative Borel function O such that O D a.e. Let Q D 1 X jD1 OjB.x;j/ 2j k OkLp.B.x;j// C 1 2 Lp .X/: If 2 , then R Q ds D 1, since is compact and thus contained in B.x; j/ for some j. Thus also Z Q n ds D 1 1 for all 2 : Hence Modp. / k Q=nkp Lp.X/ ! 0, as n ! 1. Corollary 1.38. If Modp. j / D 0 for all j, then Modp S1 jD1 j D 0. This is a special case of Lemma 1.34 (b), but actually we will only need this special case in this book. We therefore provide a simpler proof, so that the reader who so wishes can skip Lemma 1.34 (b). Proof. By Proposition 1.37, for j D 1; 2; ::: , there exists a nonnegative function j 2 Lp .X/ such that Z j ds D 1 for all 2 j : Let further D 1 X jD1 j 2j k j kLp.X/ 2 Lp .X/: Then Z ds D 1 for all 2 1 [ jD1 j : Hence Modp S1 jD1 j D 0, by Proposition 1.37. The following corollary is a direct consequence of Corollary 1.38. Note that, in the same way as upper gradients, p-weak upper gradients are not unique. Corollary 1.39. Let g and g0 be p-weak upper gradients of u and v, respectively, and a 2 R. Then jajg and gCg0 are p-weak upper gradients of au and uCv, respectively. Note that it is not true in general that g g0 is a p-weak upper gradient of u v. The following lemma strengthens the definition of p-weak upper gradients and will be often used in our proofs, even without notice.
- 36. 20 1 Newtonian spaces Lemma 1.40. If g is a p-weak upper gradient of f on X and D f 2 .X/ W g is not an upper gradient of f along g; then Modp. / D 0. Proof. Let 0 consist of those curves 0 W Œ0; l0 ! X such that jf . 0 .0// f . 0 .l0 //j 6 Z 0 g ds (which in particular is true if the integral is not defined). Then Modp. 0 / D 0 by assumption. Moreover, for every 2 there is a subcurve 0 2 0 . It follows that Modp. / D 0, by Lemma 1.34 (c). Definition 1.41. Let E D f 2 .X/ W 1 .E/ ¤ ¿g and C E D f 2 .X/ W ƒ1. 1 .E// ¤ 0g: Here ƒ1 is the Lebesgue measure on R, extended as an outer measure to all subsets of R. (Note that X D .X/.) Lemma 1.42. If .E/ D 0, then Modp. C E / D 0. Proof. Let F E be a Borel set with .F / D 0, and let D 1F . For 2 C E , we have ƒ1. 1 .F // ¤ 0. Moreover, 1 .F / is a Borel set, and thus Z ds D 1 for all 2 C E : Hence Modp. C E / Z X p d D 0: Lemma 1.43. Let g and Q g be nonnegative measurable functions on X such that g D Q g a.e. Then Z g ds D Z Q g ds for p-a.e. curve . In particular, R g ds is defined for p-a.e. curve (with a value in Œ0; 1). Proof. By Proposition 1.2 there is a nonnegative Borel function g0 which equals g a.e. Let E D fx 2 X W g.x/ ¤ g0 .x/g. As g0 is a Borel function, R g0 ds is defined for every curve . For curves 2 .X/ n C E , Z g ds D Z g0 ds: (1.5) Since .E/ D 0, we have Modp. C E / D 0, by Lemma 1.42, and thus (1.5) holds for p-a.e. curve . Similarly R Q g ds D R g0 ds for p-a.e. curve .
- 37. 1.5 p-weak upper gradients and modulus of curve families 21 Corollary 1.44. Let g be a p-weak upper gradient of f , and let Q g D g a.e., Q g 0. Then Q g is also a p-weak upper gradient of f . In particular, there is a Borel p-weak upper gradient g0 of f such that g0 D g a.e. The following example shows that the corresponding result for upper gradients is false. Example 1.45. Let f W R2 ! R be given by f ..x1; x2// D x1. Then g D 1 is an upper gradient of f . Let further g0 D gR2nR. Then g0 D g a.e. and g0 is a nonnegative Borel function which is not an upper gradient of f . Nevertheless, g0 is a p-weak upper gradient of f , by Corollary 1.44. The following result shows that the N 1;p -norm is not changed if the infimum in Definition 1.17 is taken over all p-weak upper gradients of u. Note that even though this result is mainly of interest when g 2 Lp .X/, this is not required. Lemma 1.46. Let g be a p-weak upper gradient of f . Then there exist upper gradients gj so that lim j!1 kgj gkLp.X/ D 0: (1.6) Proof. First we can find a nonnegative Borel function g0 such that g0 D g a.e. By Lemma 1.43, g0 is also a p-weak upper gradient of f . Let consist of those curves W Œ0; l ! X such that jf . .0// f . .l //j 6 Z g0 ds: By assumption, Modp. / D 0, and hence by Proposition 1.37, there is a nonnegative Borel function 2 Lp .X/ such that R ds D 1 for all 2 . Let finally gj D g0 C =j . Then gj is an upper gradient of f and (1.6) holds. The following result shows that p-weak upper gradients are a natural generalization of the (modulus of the) usual gradients. It is an improvement of Corollary 1.15 and will be further refined in Propositions A.3 and A.11. Corollary 1.47. If X D Rn and f W Rn ! R is locally Lipschitz, then jrf j (or more precisely any of its nonnegative everywhere defined representatives) is a p-weak upper gradient of f . Proof. Proposition 1.14 shows that lip f is an upper gradient of f . By the Rademacher theorem (see e.g. Theorem 2.2.1 in Ziemer [361]), f is differentiable at a.e. x 2 Rn . For such x and all y 2 B.x; r/ we have jf .y/ f .x/j D jrf .x/ .y x/ C o.jy xj/j r jrf .x/j C o.r/ and hence lip f .x/ D jrf .x/j. The result now follows from Corollary 1.44.
- 38. 22 1 Newtonian spaces We shall next see that there is an intimate relation between small sets and small curve families. Proposition 1.48. Let E X. Then Cp.E/ D 0 if and only if .E/ D Modp. E / D 0. Proof. Assume first that .E/ D Modp. E / D 0. Let u D E . Then on p-a.e. curve u 0, and therefore g 0 is a p-weak upper gradient of u. It follows that Cp.E/ kukp N 1;p.X/ kukp Lp.X/ C kgkp Lp.X/ D 0: Assume conversely that Cp.E/ D 0. It follows directly that .E/ Cp.E/ D 0. For each j D 1; 2; ::: , let uj 2 N 1;p .X/ be a nonnegative function with upper gradient gj , such that kuj kN1;p.X/ 2j ; kgj kLp.X/ 2j and uj E : Let u D P1 jD1 uj 2 Lp .X/ and g D P1 jD1 gj 2 Lp .X/. Let further F D fx 2 X W u.x/ D 1g E; D ˚ W R g ds D 1 ; .F / D f W F g: Since g 2 Lp .X/ we have Modp. / D 0. Moreover, for 2 .F / we have Z u ds D Z 1 ds D 1: As u 2 Lp .X/ we have Modp. .F // D 0, by Proposition 1.37. Let now 2 .X/n. [ .F //. Then, in particular, there is y 2 with u.y/ 1. So for all x 2 we have u.x/ D lim k!1 k X jD1 uj .x/ lim k!1 k X jD1 uj .y/ C k X jD1 juj .x/ uj .y/j u.y/ C lim k!1 k X jD1 Z gj ds D u.y/ C Z g ds 1: Thus x … F , and hence X n F . Therefore E F [ .F / has zero p-modulus. Corollary 1.49. If u D v q.e. and g is a p-weak upper gradient of u, then g is also a p-weak upper gradient of v (and thus, u and v have the same set of p-weak upper gradients).
- 39. 1.5 p-weak upper gradients and modulus of curve families 23 Proof. Let E D fx W u.x/ ¤ v.x/g. By assumption, Cp.E/ D 0, and by Proposi- tion 1.48, Modp. E / D 0. Hence u v on p-a.e. curve . It then easily follows that if g is a p-weak upper gradient of u, then it is also of v. Proposition 1.50. Assume that .E/ D 0 and that g 0 is such that for p-a.e. curve W Œ0; l ! X it is true that either .0/; .l / 2 E or ju. .0// u. .l //j Z g ds: (1.7) Then g is a p-weak upper gradient of u. The following consequence is immediate upon letting E D fx 2 X W ju.x/j D 1g. Corollary 1.51. Let uW X ! x R be a function which is finite a.e. and assume that g 0 is such that for p-a.e. curve W Œ0; l ! X it is true that either ju. .0//j D ju. .l //j D 1 or ju. .0// u. .l //j Z g ds: Then g is a p-weak upper gradient of u. The essence of this corollary is that as long as u is real-valued a.e. it makes no difference how we interpret the inequality (1.4) in Definition 1.32 in the special case when the left-hand side is either j1 1j or j.1/ .1/j. Our main interest is in N 1;p (and N 1;p loc ) functions, and such functions are necessarily real-valued a.e. Proof of Proposition 1.50. Let be the set of the exceptional curves for which (1.7) does not hold for some subcurve of . Then Modp. / D 0 by Lemma 1.34. Let also .E/ D f W Eg. Since .E/ D 0 we have Modp. .E// Modp. C E / D 0, by Lemma 1.42. Let 2 .X/ n . [ .E//. Then there is t 2 Œ0; l such that .t/ … E. If t D 0 or t D l , then ju. .0// u. .l //j Z g ds; by assumption. Otherwise ju. .0// u. .l //j ju. .0// u. .t//j C ju. .t// u. .l //j Z jŒ0;t g ds C Z jŒt;l g ds D Z g ds; since the second alternative in (1.7) holds for jŒ0;t and jŒt;l . We have thus shown that g is a p-weak upper gradient of u.
- 40. 24 1 Newtonian spaces Recall our convention that 1 1 D 1. In particular, the second alternative in (1.7) does not hold for jŒ0;t if t D 0 and u. .0// D ˙1, and therefore the cases t D 0 and t D l needed separate treatment above. Note that in the context of Corollary 1.51 this is not necessary. Lemma 1.28 has a direct counterpart for p-weak upper gradients. (Assuming that gi are merely p-weak upper gradients, we obtain a p-weak upper gradient g with the same proof.) However, using Proposition 1.50 (with E D fx 2 X W u.x/ D 1g), we can show the following stronger result. Lemma 1.52. Letuj befunctionswithp-weakuppergradientsgj , andletu D supj uj , g D supj gj and E D fx W u.x/ D 1g. If .E/ D 0, then g is a p-weak upper gradient of u. Note that unlike in Lemma 1.28, here we do not assume that uj are uniformly bounded from above. Recall also the discussion in Remark 1.29 showing that without any condition on E this lemma would be false. Proof. Let j be the set of the exceptional curves along which gj is not an upper gradient of uj , j D 1; 2; ::: . Let W Œ0; l ! X be a curve not in S1 jD1 j and let x D .0/ and y D .l /. Assume that either u.x/ 1 or u.y/ 1. Then, as in the proof of Lemma 1.28, ju.x/ u.y/j sup j juj .x/ uj .y/j sup j Z gj ds Z g ds: It follows from Corollary 1.51 that g is a p-weak upper gradient of u. Finally, we show that for almost all purposes we can forget about X n supp and replace X by supp . Note, however, that we do not want to assume X D supp in general, see the comment before Proposition 1.6. Proposition 1.53. Let Z D X n supp . Then Cp.Z/ D 0 and Modp. Z/ D 0. Proof. As Z is open, the function g D 1Z is an upper gradient of u D Z as R g ds D 1 for any curve starting in Z and ending in supp . Hence Cp.Z/ kukp N1;p.X/ D 0. That also Modp. Z/ D 0 follows from Proposition 1.48. 1.6 Banach space and ACCp On Rn it is well known that every Sobolev function has a representative which is ACL, i.e. absolutely continuous on almost every line parallel to the axes, see e.g. Theorem 2.1.4 in Ziemer [361]. On metric spaces we have no preferred lines, but we can obtain a stronger result, Theorem 1.56 below. In fact, for this we only need the existence of a p-integrable upper gradient, which justifies the following definition.
- 41. 1.6 Banach space and ACCp 25 Definition 1.54. We say that a measurable function belongs to the Dirichlet space Dp .X/ if it has an upper gradient in Lp .X/. In view of Lemma 1.46 it is equivalent to assume the existence of a p-weak upper gradient in Lp .X/. Clearly, N 1;p .X/ Dp .X/, and Lemma 1.16 immediately implies that Dp .X/ is a vector space. Theorem 1.55. The space Dp .X/ is a lattice. Proof. The proof of Theorem 1.20 in fact also yields this result. Theorem 1.56. If u 2 Dp .X/, then u 2 ACCp.X/, i.e. u is absolutely continuous on p-a.e. curve in the sense that u ı W Œ0; l ! R is absolutely continuous for p-a.e. curve in X. Let us recall the definition of absolute continuity. Definition 1.57. A function f W Œa; b ! R is absolutely continuous on Œa; b if for every 0 there is ı 0 such that n X iD1 jf .bi / f .ai /j for any n and any a a1 b1 a2 b2 an bn b such that n X iD1 .bi ai / ı: This is not the right place to study the basic facts about absolutely continuous functions. Let us however recall that a large motivation for their study is the fact that f is absolutely continuous on Œa; b if and only if f 0 2 L1 .Œa; b/ and f .x/ D f .a/ C Z x a f 0 .t/ dt for all x 2 Œa; b: See e.g. Folland [124], Theorem 3.35, or Rudin [311], Theorem 7.20 together with the comments just before Definition 7.17. We will need the following simple facts. Lemma 1.58. If u and v are absolutely continuous on Œa; b, then u C v, maxfu; vg and uv are also absolutely continuous on Œa; b. If uW Œa; b ! Œc; d is absolutely continuous and v W Œc; d ! R is Lipschitz, then v ı u is also absolutely continuous on Œa; b.
- 42. 26 1 Newtonian spaces Proof. Let us start with uv: As u and v obviously are continuous they are bounded on Œa; b, say by M 0. Let 0 and choose ı 0 so small that if Pn iD1.bi ai / ı for any a a1 b1 a2 b2 an bn b, then n X iD1 ju.bi / u.ai /j 2M and n X iD1 jv.bi / v.ai /j 2M : Thus n X iD1 ju.bi /v.bi / u.ai /v.ai /j n X iD1 .ju.bi /v.bi / u.bi /v.ai /j C ju.bi /v.ai / u.ai /v.ai /j/ D n X iD1 .ju.bi /j jv.bi / v.ai /j C jv.ai /j ju.bi / u.ai /j/ M n X iD1 jv.bi / v.ai /j C M n X iD1 ju.bi / u.ai /j M 2M C M 2M D : To show that u C v and maxfu; vg are absolutely continuous is easier and we leave it to the reader. As for the composition v ı u, let 0 and choose ı according to the definition of absolute continuity of u. Let also M 0 be a Lipschitz constant for v. Then n X iD1 j.v ı u/.bi / .v ı u/.ai /j M n X iD1 ju.bi / u.ai /j M for any n and any a a1 b1 a2 b2 an bn b such that n X iD1 .bi ai / ı: Proof of Theorem 1.56. By Definition 1.54, there is an upper gradient g 2 Lp .X/ of u. Let be the collection of all curves in X such that R g ds D 1. Then Modp. / D 0 by Proposition 1.37. Let now 2 .X/ n . Then for all a; b 2 Œ0; l , j.u ı /.a/ .u ı /.b/j Z jŒa;b g ds 1; (1.8) and, in particular, .u ı /.a/; .u ı /.b/ 2 R.
- 43. 1.6 Banach space and ACCp 27 Assume that u is not absolutely continuous on , i.e. that f WD uı is not absolutely continuous on Œ0; l . Then there is an 0 such that for every j D 1; 2; ::: , there are 0 aj;1 bj;1 aj;nj bj;nj l such that nj X iD1 .bj;i aj;i / 1 2j and nj X iD1 jf .bj;i / f .aj;i /j : Let Ej D Snj iD1Œaj;i ; bj;i . Then by (1.8) and dominated convergence, nj X iD1 jf .bj;i / f .aj;i /j Z jEj g ds ! 0; as j ! 1: But this is a contradiction. Hence f is absolutely continuous on , and u is absolutely continuous on p-a.e. curve. We shall next see that the equivalence classes for Newtonian functions are finer than for Sobolev functions. Remember that functions in N 1;p .X/ are defined everywhere and that z N1;p .X/ is the corresponding quotient space, see Definition 1.19. Proposition 1.59. If u; v 2 ACCp.X/ (in particular if u; v 2 N 1;p .X/), and u D v a.e., then u D v q.e. Proof. Without loss of generality we may assume that v 0 (consider otherwise u v). Let E D fx W u.x/ ¤ 0g. Since .E/ D 0, we have Modp. C E / D 0. We also know that u is absolutely continuous on p-a.e. curve. So on p-a.e. curve , u is both absolutely continuous and ƒ1. 1 .E// D 0. On such curves u D 0 a.e. with respect to arc length, and by continuity everywhere. Thus … E . Hence Modp. E / D 0, and by Proposition 1.48, Cp.E/ D 0. Corollary 1.60. If u; v 2 ACCp.X/ (in particular if u; v 2 N 1;p .X/), and u v a.e., then u v q.e. Proof. Let f D minfu; vg 2 ACCp.X/. Since u v a.e. we have f D v a.e. Thus u f D v q.e., by Proposition 1.59. The following result shows that the equivalence classes in z N 1;p .X/ are up to sets of capacity zero, and not up to measure zero as for usual Sobolev spaces on Rn . This also shows that two functions in the same equivalence class in z N1;p .X/ have the same set of p-weak upper gradients (by Corollary 1.49). Proposition 1.61. Let uW X ! x R be a function. Then kukN1;p.X/ D 0 if and only if Cp.fx W u.x/ ¤ 0g/ D 0. Moreover, if u 2 N 1;p .X/ and v W X ! x R, then v D u q.e. if and only if v 2 N 1;p .X/ and v u.
- 44. 28 1 Newtonian spaces Proof. Let E D fx W u.x/ ¤ 0g. Assume first that kukN1;p.X/ D 0. Then u 2 N 1;p .X/ and u D 0 a.e., so by Proposition 1.59, u D 0 q.e., and thus Cp.E/ D 0. Conversely, assume that Cp.E/ D 0. Corollary 1.49 shows that 0 is a p-weak upper gradient of u. Moreover, .E/ D 0, by Theorem 1.27. Hence kukN1;p.X/ D 0. The last part is obtained by applying the first part to u v. We are now ready to show that the norm in y N 1;p .X/, which was introduced after Definition 1.19, is well defined. Lemma 1.62. Assume that u 2 y N 1;p .X/ and that v; w 2 N 1;p .X/ are such that u D v D w a.e. Then kvkN1;p.X/ D kwkN 1;p.X/, i.e. kuk y N1;p.X/ is well defined and equal to kvkN 1;p.X/ D kwkN1;p.X/. Proof. It follows from Proposition 1.59 that v D w q.e. Hence, by Proposition 1.61, kv wkN 1;p.X/ D 0, which implies that kvkN1;p.X/ D kwkN 1;p.X/. As there are sets with zero measure and positive capacity we have shown that N 1;p .X/ and y N 1;p .X/ are (in general) different spaces. The following results give some more insight in this direction. Proposition 1.63. Let u 2 y N 1;p .X/. Then u 2 N 1;p .X/ if and only if u 2 ACCp.X/. Proof. The necessity follows from Theorem 1.56. For the sufficiency assume that u 2 ACCp.X/. By assumption, there is v 2 N 1;p .X/ ACCp.X/ such that u D v a.e., but then u D v q.e., by Proposition 1.59. Hence, by Proposition 1.61, kuvkN1;p.X/ D 0, and thus also u 2 N 1;p .X/. Example 1.64. The function R equals zero a.e. in R2 and is therefore a representative of a Sobolev function. It is, however, a bad representative of the zero function, as it is not absolutely continuous on any line parallel to the y-axis. In our notation, it belongs to y N 1;p .R2 /, but not to N 1;p .R2 /. In view of Proposition 1.61, any Newtonian representative of the zero function must be zero q.e. Proposition 1.65. y N 1;p .X/= ae Š z N 1;p .X/ WD N 1;p .X/= , where u ae v if and only if u D v a.e. In Rn and many other situations, Sobolev spaces are usually defined as sets of a.e.- equivalence classes, which corresponds to the left-hand side above. This result shows that we obtain essentially the same Sobolev spaces with our definition, even though we insist on having the smaller q.e.-equivalence classes. We have just weeded out the bad representatives of Sobolev functions, see Example 1.64. Our q.e.-representatives have the additional property of belonging to ACCp.X/, something not shared by general a.e.-representatives. We will later find additional advantages of q.e.-representatives, e.g. in connection with quasicontinuity (see e.g. Proposition 5.33) and resolutivity (see e.g. Theorem 10.12).
- 45. 1.6 Banach space and ACCp 29 Proof. Let f1 and f2 be two representatives of one equivalence class in z N 1;p .X/. Then f1; f2 2 N 1;p .X/ y N 1;p .X/. Moreover, f1 D f2 q.e., by Proposition 1.61, and hence a.e., by Theorem 1.27. Thus they belong to the same equivalence class in y N 1;p .X/= ae. Conversely, let f1 and f2 be two representatives of one equivalence class in y N 1;p .X/= ae. Then, by the definition of y N 1;p .X/, there are f 0 1; f 0 2 2 N 1;p .X/ such that f 0 j D fj a.e., j D 1; 2. Hence also f 0 1 D f 0 2 a.e. and consequently, q.e., by Proposition 1.59. Therefore f 0 1 and f 0 2 both define the same equivalence class in z N 1;p .X/, and there is thus a one-to-one correspondence between the a.e.-equivalence classes in y N 1;p .X/= ae and the q.e.-equivalence classes in z N 1;p .X/. Finally, kf1k y N 1;p.X/ D kf 0 1kN1;p.X/, showing that the norms are the same for the corresponding equivalence classes. Proposition 1.63 gave us a characterization of “good” representatives, but it is not obvious from this characterization that a continuous representative is always “good”. Let us therefore show this, and a bit more. Proposition 1.66. Let u 2 y N1;p .X/ and E X be such that Cp.E/ D 0 and ujXnE is continuous. Then u 2 N 1;p .X/. Note that we do not require u to be continuous at all points in X n E, we only require that the restriction of u to X n E is continuous. A function u satisfying our condition is weakly quasicontinuous, see Definition 5.17 (but the converse is not true). In fact, (weak) quasicontinuity characterizes the “good” representatives, under some additional assumptions on X, see Proposition 5.33. We do not know if that characterization holds in general, and therefore the result here is of interest. Proof. By assumption, there is a function v 2 N1;p .X/ such that v D u a.e. Let A D fx W u.x/ ¤ v.x/g so that .A/ D 0. By Lemma 1.42 and Proposition 1.48, Modp. E / D Modp. C A / D 0. As v 2 ACCp.X/, by Theorem 1.56, we thus see that p-a.e. curve is such that v is absolutely continuous on and … E [ C A . Since … E , u is continuous on . Moreover, as … C A , u D v a.e. on . As u and v are continuous on and equal a.e. on , we must have u D v everywhere on . In particular, u is absolutely continuous on , and hence u 2 ACCp.X/. By Proposition 1.59, u D v q.e., and thus u 2 N 1;p .X/, by Proposition 1.61. A corresponding result is true also for Dp in the following form. Proposition 1.67. Let uW X ! x R be such that there is v 2 Dp .X/ with u D v a.e. Let further E X be such that Cp.E/ D 0 and ujXnE is continuous. Then u 2 Dp .X/. The proof is the same as above, apart from that rather than applying Proposition 1.61 at the very end, we need to apply its proof. Another consequence of our definition of Newtonian functions is the equality be- tween the essential supremum and the q.e.-essential supremum defined as follows.
- 46. 30 1 Newtonian spaces Definition 1.68. For A X and f W A ! x R, define Cp- ess sup A f D inffk 2 R W Cp.fx 2 A W f .x/ kg/ D 0g; Cp- ess inf A f D supfk 2 R W Cp.fx 2 A W f .x/ kg/ D 0g: As a corollary of Proposition 1.59 we make the following useful observation. Corollary 1.69. If u 2 ACCp.X/ (in particular if u 2 N 1;p .X/), then ess sup X u D Cp- ess sup X u and ess inf X u D Cp- ess inf X u: Proof. Theorem 1.27 (ii), shows that ess supX u Cp- ess supX u. In particular, the equality holds if ess supX u D 1. Otherwise, let ess supX u, 2 R, be arbitrary. Then u a.e., and thus u q.e., by Corollary 1.60, showing that Cp- ess supX u . Letting ! ess supX u shows that ess supX u Cp- ess supX u. The second equality follows by applying the first equality to u. The following generalization of Proposition 1.30 is also worth pointing out. Corollary 1.70. If u 2 ACCp.X/ and u is finite a.e., which in particular holds if u 2 N 1;p .X/, then u is finite q.e. Proof. Let E D fx W u.x/ 2 Rg and let v D uE . Let be a curve on which u is absolutely continuous. Then v D u on . Thus v 2 ACCp.X/. By Proposition 1.59, v D u q.e., and as v is real-valued the conclusion follows. Theorem 1.71. The space z N 1;p .X/ D N 1;p .X/= is a Banach space. Proof. That z N1;p .X/ is a normed linear space was observed after Definition 1.19. We need to show the completeness. Let fuj g1 jD1 be a Cauchy sequence in N 1;p .X/. By passing to a subsequence, if necessary, we can assume that kujC1 uj kN1;p.X/ 2j.pC1/=p 2j : (1.9) Let Ej D fx 2 X W jujC1.x/ uj .x/j 2j g: Then Cp.Ej / 2jp kujC1 uj kp N1;p.X/ 2j . (Note that if jujC1.x/j D juj .x/j D 1, then we consider jujC1.x/ uj .x/j D 1, and thus x 2 Ej . In view of Proposi- tion 1.30 this does not cause any trouble.) Let Fk D 1 [ jDk Ej and F D 1 kD1 Fk:
- 47. 1.7 Examples 31 Then Cp.Fk/ 21k and Cp.F / D 0. Let x 2 X n F . Then x 2 X n Fl for some l and jujC1.x/ uj .x/j 2j for all j l. Hence uj .x/ is a Cauchy sequence in R and we can define u.x/ D lim j!1 uj .x/ D uk.x/ C 1 X jDk .ujC1.x/ uj .x//: (1.10) The function u is defined q.e. (and we may define it arbitrarily elsewhere) and ku ukkLp.X/ 21k , by (1.9). As Cp.F / D 0, p-a.e. curve in X has empty intersection with F , by Proposition 1.48. Let be one such curve, connecting x and y. Then by (1.10), j.uuk/.x/.uuk/.y/j 1 X jDk j.ujC1 uj /.x/.ujC1 uj /.y/j Z 1 X jDk gj ds; where gj is an upper gradient of ujC1 uj such that kgj kLp.X/ 2j . Hence, Q gk D P1 jDk gj is a p-weak upper gradient of u uk and k Q gkkLp.X/ 21k . It follows that ku ukkN1;p.X/ ! 0, as k ! 1. As a corollary of the proof we obtain the following result. Corollary 1.72. Assume that uj ! u in N 1;p .X/, as j ! 1. Then there is a subsequence which converges to u pointwise q.e. Moreover, for every 0 there is a set E with Cp.E/ , such that the subsequence converges uniformly to u outside of E. Furthermore, if all uj are continuous, then the subsequence converges uniformly (though not necessarily to u) outside open sets of arbitrarily small capacity. Proof. In the notation of the proof of Theorem 1.71, we obtain a subsequence (again denoted by fuj g1 jD1) which converges to Q u 2 N 1;p .X/ uniformly outside of Fk for every k, and thus pointwise outside of F . Note that if all uj are continuous, then all Fk are open. Clearly, u Q u and Proposition 1.61 shows that u D Q u q.e. Letting E D Fk [ fx W u.x/ ¤ Q u.x/g, for suitably large k, finishes the proof. 1.7 Examples Let us take the opportunity to mention some examples of metric spaces on which the Newtonian theory can be interesting. (See also Appendix A, where examples of spaces with doubling measures supporting Poincaré inequalities are discussed in greater detail.) The first example is of course Rn . In this case the Newtonian space N 1;p .Rn / is essentially the standard Sobolev space W 1;p .Rn /. More precisely, if Rn , then
- 48. 32 1 Newtonian spaces W 1;p ./ D y N 1;p ./ and N 1;p ./ is the same space but with less representatives in the equivalence classes, in fact it singles out exactly the nice representatives which are quasicontinuous, as we shall see in Section 5.2. The same is true on weighted Rn with p-admissible weights, see Appendix A.2. The next type of examples that comes to mind are probably Riemannian manifolds. Again, the Newtonian space will be the same as the usual Sobolev space, but with only the good, quasicontinuous, representatives. A particular application here is that one can take limits of sequences of Riemannian manifolds, under Gromov–Hausdorff con- vergence (which is sometimes called Vietoris convergence). Such limits are typically not manifolds, but are indeed metric spaces and Newtonian spaces are well suited to be defined on them. Another example is the Heisenberg group H1 as well as higher order Heisen- berg groups, more general Carnot groups and Carnot–Carathéodory spaces, see Ap- pendix A.6. Yet other examples are simplicial complexes, with simplices glued to each other. Such glueing can be made along different types of faces and it is even possible for the complex to have (highest-order) simplices of different dimension in different parts. The simplest example is probably glueing the segment Œ1; 0 f0g (with the one- dimensional Lebesgue measure) to the triangle f.x; y/ W 0 y x 1g (with a weighted two-dimensional measure) at .0; 0/. SeeAppendixA.4 for more complicated examples. Even discrete objects, such as graphs, can be included in our theory. Here one has to consider the so-called metric graphs, see Appendix A.5. All of the examples above can be equipped with doubling measures and support Poincaré inequalities, under suitable conditions. So in particular, most of the results we show in Chapters 3–6 are valid for these spaces, and they are also under consideration for the potential theory developed in Chapters 7–14. In many of these examples one has a natural vector-valued gradient, and one can also apply Cheeger’s theorem (Theo- rem B.6) to show that a vector-valued gradient structure exists. A disadvantage of using Cheeger’s theorem is that it does produce a vector-valued gradient structure, but this structure is not unique. Moreover, one always uses an inner product to take the length of the Cheeger gradient. It is thus not easy to understand what the Cheeger gradient, or its length, really determines, in particular its connection with the geometry of the space. The upper gradient on the other hand has a very geometric definition. Nevertheless, the results we obtain in Chapters 7–14 are valid also for Cheeger p-harmonic functions, see Section 7.1 and Appendix B.2. Further examples include open subsets of the spaces above, as well as closed and other nonopen subsets. Such examples usually do not support Poincaré inequalities, nor are the restricted measures usually doubling, see however Appendix A.3, where uniform domains are discussed. For open subsets one still gets a natural gradient from the gradient in the ambient space, a fact that we exploit to deduce many of our results, such as quasicontinuity, for open subsets of (often complete) doubling p-Poincaré spaces.
- 49. 1.7 Examples 33 For nonopen sets one cannot just restrict the gradient from the ambient space. Here comes another advantage of the Newtonian approach, the Newtonian spaces are directly definable on nonopen subsets of our ambient space. For other types of Sobolev spaces on X one can of course say that a function f is in the Sobolev space on a subset E if there is some open set G E such that f is in the Sobolev space on G (with G being allowed to depend on f ). For Newtonian spaces the advantage is that we do not have to go outside of E to define N 1;p .E/. So, in particular, the Newtonian approach is ideal for defining Sobolev spaces on various closed subsets E of Rn . Of course, one has to remember that if E contains too few curves then N1;p .E/ may be of little use, as e.g. with the von Koch snowflake curve, see Example 1.23. However, in the case when E is the closure of an open set, then the Newtonian space N 1;p .E/ has applications in the boundary value theory, and does make results more general than what can be achieved using classical Sobolev spaces. In such cases we do not know if all functions in N 1;p .E/ are quasicontinuous, but nevertheless they are better representatives than the usual a.e.-Sobolev functions, see the discussion on the zero p-weak upper gradient property in the notes to Chapter 5. There is yet another reason for studying potential theory on subsets of Rn (or other metric spaces). Consider X D R2 n ..1; 1/ .0; 1// which is a complete doubling p-Poincaré space. The potential theory we develop in the later part of the book can thus be applied. Let D B..0; 0/; 1/ be the unit ball. In X, it has the boundary @X D f.x; y/ 2 R2 W x2 C y2 D 1 and y 0g. Let also E D .1; 1/ f0g and f 2 C.@X /. By the methods in Chapter 10 we can solve the Dirichlet problem and find the p-harmonic function Pf in having boundary values f on @X . One can actually show that this solution is the p-harmonic function u D Pf in n E having boundary values f on @X and satisfying the zero Neumann boundary condition @u @n D 0 on E; see also Example 8.18. This is true in much more general situations. When the missing boundary, denoted by E above, is less regular, we often cannot talk about normal derivatives in the usual sense. Our way of solving the Dirichlet problem in such situations still gives rise to a very general weak-type zero Neumann boundary condition. Theorem A.21 shows that the closure of any uniform domain in Rn serves as a complete doubling p-Poincaré space and thus the above mixed boundary value problem can be considered on subsets of it. For instance, we can let X be the closed bounded set in R2 , whose boundary is the von Koch snowflake curve K, see Example 1.23. Solving the Dirichlet problem on an open subset of X corresponds to solving the mixed boundary value problem on n K with Dirichlet boundary values on @X and zero Neumann boundary values on K . Thus we are actually able to handle rather general mixed boundary value problems within our treatment of the Dirichlet problem.
- 50. 34 1 Newtonian spaces 1.8 Notes Upper gradients were introduced under the name very weak gradients by Heinonen and Koskela in [173], [174]. They were renamed upper gradients, for obvious reasons, when Koskela and MacManus [229] studied p-weak upper gradients for the first time. In Semmes [314], Definition 1.8, upper gradients are called generalized gradients. Cheeger [91] defines generalized upper gradients, which for p 1 are roughly the same as p-weak upper gradients. Cheeger also defines Sobolev spaces on general metric spaces that coincide with y N 1;p .X/ if p 1, see Appendix B.2. In some of the early papers on upper and p-weak upper gradients, it is not stated explicitly if the (p-weak) upper gradients are required to be Borel functions or not. However, since Borelness is sometimes used it may have been implicitly assumed. Requiring that upper gradients be Borel functions has some definite advantages. First of all, the concept of upper gradients becomes totally independent of the measure, and second, the curve integrals involved are always defined. As for p-weak upper gradients the situation is different. At first (when care was taken) it was assumed that p-weak upper gradients be Borel, the reasons for this being that it would prevent any problems with the curve integrals not being defined, and in view of Proposition 1.2 it was not believed to do any harm. However, when working on the book Heinonen–Koskela–Shanmugalingam–Tyson [177], it was observed by Heinonen (as communicated to us by Shanmugalingam [322]), that care has to be taken in glueing formulas such as in Section 2.4, as the glued p-weak upper gradient may fail to be Borel even if the initial p-weak upper gradients are Borel. The immediate response was that one has to take extra care when formulating similar results. In Björn–Björn [44], Section 3 (which incidentally does not appear in Björn– Björn [45]), a different remedy was suggested which we follow here: to omit the assumption of Borelness for p-weak upper gradients. Let us name some of the rea- sons for this: (a) In Definition 1.32 it is enough if the curve integrals are defined for p-a.e. curve, and in fact this is automatic for nonnegative measurable functions, by Lemma 1.43. (b) Just requiring measurability of course means that the concept of p-weak upper gradients, in contrast to that of upper gradients, is dependent on the measure. However, even the concept of Borel p-weak upper gradients is dependent on the measure as the inequality is required to just hold for p-a.e. curve. (c) The glueing formulas in Section 2.4 become more appealing. Also, how to interpret j1 1j is often not mentioned explicitly. However, tak- ing into account Corollary 1.51, it makes no difference for p-weak upper gradients how j1 1j is interpreted, at least not when only Newtonian functions are under consideration. Lemma 1.34 for metric spaces is in Heinonen–Koskela [174], p. 9, where they observe that the proofs of the corresponding Euclidean results in Fuglede [128] carry over verbatim. Fuglede was the first one to study p-modulus. The 2-modulus and its inverse extremal length were introduced by Beurling in the early 1930s, but first published in the joint papersAhlfors–Beurling [6], [7]. Proposition 1.37 also goes back
- 51. 1.8 Notes 35 to Fuglede [128]. A thorough study of extremal length and p-modulus for p 1 in weighted Rn is in Ohtsuka [299]. Newtonian spaces were first defined in Shanmugalingam [318] and [319]. Therein one can find Theorems 1.20 and 1.71, Corollaries 1.21 and 1.72, and Lemma 1.42. Also Theorem 1.56 and Proposition 1.59 (stated only for u 2 N1;p .X/) can be found in [319], as well as the “only if” part of Proposition 1.48. The “if” part seems to be new here. An equivalence between the modulus and a variational capacity (see Section 6.3) has been obtained by Heinonen–Koskela [174] and Kallunki [Rogovin]– Shanmugalingam [194]. Some estimates for the modulus of curve families in ring domains on metric spaces can be found inAdamowicz–Shanmugalingam [3] and Garo- falo–Marola [138]. Lemma 1.46 was first proved by Koskela–MacManus [229], Lemma 2.4. Lem- ma 1.43 is from Björn–Björn [44] (but is not included in [45]), while Corollary 1.51 is from Björn–Björn–Parviainen [54]. The direction in Proposition 1.63 which is not covered by Theorem 1.56, as well as Propositions 1.66 and 1.67, seems to be new here. If X is a complete doubling p-Poincaré space, then any function with an upper gradient in Lp .X/ is measurable, byTheorem 4.52. Thus the measurability assumption can be dropped from the definition of the Dirichlet space Dp .X/, Definition 1.54, under these assumptions. In general, however, it is easy to see that measurability is not a redundant assumption in Definition 1.54, consider e.g. the von Koch snowflake curve in Example 1.23. For p D 1, one often studies BV-functions, functions of bounded variation, which in some problems appear as the natural limit of W 1;p , as p ! 1C. We have not pursued that, since our main interest is to study the theory for p 1 and we in- clude the case p D 1 only when it comes for free. BV-functions have been studied on metric spaces by Ambrosio [14], [15], Camfield [83], Miranda [286], Ambrosio– Miranda–Pallara [16], Kinnunen–Korte–Shanmugalingam–Tuominen [210], [211], Hakkarainen–Kinnunen [156] and Hakkarainen–Shanmugalingam [157]. A further generalization of Newtonian spaces are Orlicz–Sobolev spaces based on upper gradients. They have been studied by e.g. Aïssaoui [12], [13], Heikkinen– Tuominen [167], Mocanu [291] and Tuominen [339], [340], [341]. Estimates and in- equalities for Orlicz–Sobolev capacities on metric spaces can be found inAïssaoui [11], J. Björn–Onninen [71], Costea [102] and Tuominen [339]. Mocanu [288], [289], [290] studies Newtonian spaces and capacities in even more general settings. Variable ex- ponent Newtonian spaces N1;p.x/ were studied by Harjulehto–Hästö–Pere [160] and Futamura–Harjulehto–Hästö–Mizuta–Shimomura [129]. Throughout the book we have restricted ourselves to real-valued Newtonian func- tions. Newtonian spaces for Banach-space valued functions were studied by Heino- nen–Koskela–Shanmugalingam–Tyson [176], [177]. The (first part of the) 5-covering lemma (Lemma 1.7) has many names in the literature, including the basic covering lemma and the simple Vitali lemma and is often attributed to Vitali [348] or Wiener [353]. However, Vitali [348] obtains a much stronger result under stronger assumptions: he requires every point in the set E under
- 52. 36 1 Newtonian spaces consideration to be covered by arbitrarily small balls (a so-called Vitali covering) and obtains a disjoint covering of all of E but for a set of measure zero. (Vitali’s formulation is a bit different, but it is relatively easy to see that it is equivalent to the modern formulation, especially for open E.) It is therefore more correct to reserve the name Vitali covering theorem for this result. Wiener [353], on the other hand, may have been the first to consider coverings which need not be arbitrarily small around every point. He assumes that every point in E is the centre of some ball in the covering and finds a disjoint subfamily whose measure is at least a fixed positive portion of the measure of E. However, he does not consider blow-ups of the disjoint balls (not even in the proof). Morse [292], Theorem 3.5, seems to be the first one proving the 5-covering lemma as formulated here, and he does so in metric spaces. In fact, Morse studies coverings by more general sets and the 5-covering lemma is a special case of his result. This may have been the first time when the conclusion is that certain blow-ups of the disjoint sets (balls in our case) cover E, rather than a measure-theoretic conclusion as in Vitali’s and Wiener’s theorems. However, already Banach [27], when giving a new proof of (the modern formulation of) Vitali’s covering theorem, shows that certain blow-ups of the disjoint balls cover E, but he does not obtain the 5-covering lemma since he has stronger assumptions. He too studies coverings by more general sets than balls. Federer [123], Theorem 2.8.4, proves an even more general result than Morse, from which the 5-covering lemma follows by choosing D 2 and letting ı be the radius of the balls. Choosing D 1 C 1 2 gives a .3 C /-covering lemma, cf. Remark 1.8. Let us finally mention that the name Newtonian comes from the fact that the def- inition of upper gradients is related to the Newton–Leibniz formula, and the notation L1;p was already reserved for other spaces.
- 53. Chapter 2 Minimal p-weak upper gradients By definition, upper gradients are not unique. Nor are the p-weak upper gradients. Indeed, adding a nonnegative Borel (measurable) function to a (p-weak) upper gradient results in a new (p-weak) upper gradient. Later in this book we will study p-harmonic functions as local minimizers of the p-energy integral inf g Z gp d (2.1) from the definition of k kN 1;p.X/. It is not obvious that this infimum can be at- tained. Example 1.31 shows that in general this is impossible for upper gradients, but fortunately, the situation is different for p-weak upper gradients. In this chapter, we shall prove the existence and study some properties of minimal p-weakuppergradients, whichminimize(2.1)andaretruesubstitutesforjrujinmetric spaces. Note however that unless there are some additional geometrical assumptions on the metric space, the minimal p-weak upper gradient need not provide any control of the function itself, cf. Example 1.22. 2.1 Fuglede’s lemma We start this section by proving a fundamental lemma which guarantees that a mini- mizing sequence in (2.1) gives rise to a p-weak upper gradient. We also provide several useful consequences of this lemma, which will be needed later. Lemma 2.1 (Fuglede’s lemma). Assume that gj ! g in Lp .X/, as j ! 1. Then there is a subsequence (again denoted by fgj g1 jD1) such that for p-a.e. curve , Z gj ds ! Z g ds; as j ! 1; where all the integrals are well defined and real-valued. Moreover, for p-a.e. curve , Z jgj gj ds ! 0; as j ! 1: In fact, we will only use this with g and gj nonnegative. However, it may be worth observing that the positivity of g and gj are not essential for this result. The second part is an easy consequence of the first part, but will actually not be needed here.
- 54. 38 2 Minimal p-weak upper gradients Proof. By passing to a subsequence we may assume that kgj gkLp.X/ 2j : By Lemma 1.43, R gC ds is well defined with a value in Œ0; 1 for p-a.e. curve. By Proposition 1.37, R gC ds 1 for p-a.e. curve. Similarly R g ds is well defined and real-valued for p-a.e. curve. Thus R g ds is well defined and real-valued for p-a.e. curve. Arguing similarly for each j and using Corollary 1.38, we see that there is a curve family with Modp. / D 0 and such that the integrals R gj ds, j D 1; 2; ::: , and R g ds are well defined and real-valued for 2 .X/ n . Let next z D ² 2 .X/ n W Z gj ds 6! Z g ds; as j ! 1 ³ ; k D ² 2 .X/ n W lim sup j!1 Z jgj gj ds 1 k ³ ; k D 1; 2; ::: : It is easy to see that z S1 kD1 k, and that it is enough to show that Modp. k/ D 0 for every k. Let m D k P1 jDmC1 jgj gj. Then Z m ds 1 for all 2 k and m D 1; 2; ::: : So Modp. k/ k mkp Lp.X/ .k2m /p ! 0, as m ! 1. The last part follows directly as Modp. k/ D 0 for every k, but can also be deduced by applying the first part to the functions g0 j D jgj gj and g0 D 0. For us, the most important application of Fuglede’s lemma will be the following convergence results for p-weak upper gradients. For future references, we formulate them as three propositions, even though they are closely related. In particular, Propo- sition 2.2 is almost a special case of Proposition 2.4, but the assumptions on f are weaker and the proof is simpler. In fact, as we mainly work with Newtonian functions, Proposition 2.4 is not needed later in the book, but we have chosen to include it for completeness and future references. It combines some features of Propositions 2.2 and 2.3. Note also that in view of Example 1.31 there is no hope for similar convergence results for upper gradients. Proposition 2.2. Assume that f 2 Dp .X/ and that gj 2 Lp .X/ are p-weak upper gradients of f , j D 1; 2; :::. Assume further that gj ! g in Lp .X/, as j ! 1, and that g is nonnegative. Then g is a p-weak upper gradient of f . Proof. By Lemma 1.40 and Fuglede’s lemma (Lemma 2.1), p-a.e. curve is such that gj is an upper gradient of f along for all j D 1; 2; ::: , and R gj ds ! R g ds 2 R, as j ! 1. Consider such a curve W Œ0; l ! X. Then jf . .l // f . .0//j lim j!1 Z gj ds D Z g ds:
- 55. Other documents randomly have different content
- 56. Maud. Are you sure they are all-wool? This piece feels rather harsh to me. Newcome. Every thread, madam; that I will guarantee. We are not allowed to misrepresent anything in this establishment. You can see for yourself. [Recklessly frays out a few inches of the brown. Ethel (also fingering goods). Yes, they are all-wool; French, did you say? Newcome. Every piece imported. We keep no domestic woollen goods whatever. We have no call for anything but the foreign goods. Maud. How wide did you say? Newcome. Double width, madam—forty-four inches. Ethel. Five, seven—let me see, it would take about—how much do you usually sell for a costume? Newcome (with hilarity, holding up the browns). From eight to ten yards, madam, according to the size of the lady. For your size I should say eight yards was an abundance—a great abundance. Ethel. She is just about my size, isn't she, Maud? Maud. Just about. It wouldn't take eight yards, I shouldn't think, of such wide goods made in Empire style. Ethel. No, I suppose not; but then it's always nice to have a piece left over for new sleeves, you know. Maud. Yes, that's so. Newcome. An elegant shade, ladies, becoming to anyone, fair or dark. I am sure any lady must be pleased with a dress off of one of these —serviceable, stylish, the height of fashion. Ethel. Is brown really so fashionable this season?
- 57. Newcome. I am sure we have sold a thousand yards of these browns to ten of any other color. Maud. Is that so? Ethel. I do wonder if she really would prefer brown. What do you think, dear? Maud. Well, it depends somewhat, I think, on how she is going to have it made. Ethel. True. Well, I think she said in directoire. Maud. Plain full skirt? Ethel. Yes, smocked all around—no drapery at all. Maud. Candidly, love, do you like a skirt without any drapery at all? Ethel. Well, no, I can't say I do. Do you? Maud. No. I like a little right in the back, you know—not too much. But I think a little takes off that dreadfully plain look. Don't you? Ethel. Yes. Maud. How are y— I mean how is she going to have the waist? Ethel. I don't know. I heard her say that she was going to have a puff on the sleeve. Maud. At the elbow? Ethel. No, at the shoulder. Maud. And revers, I suppose. Ethel. Yes, those stylish broad ones. Maud. Of velvet? Ethel. Velvet or plush.
- 58. Newcome (who has been manfully holding the browns up above his head, permits them to gently descend). We have a full line in plushes and velvets, ladies, to match all these shades. Maud. How nice! Ethel. So convenient! Newcome (mildly). Do you think you'll decide on the brown, madam? Ethel. Oh, dear! I don't know. It is so hard to shop for some one else! Maud. It is horrid. Ethel. I vow every time I do it that it shall be the last. I am always so afraid of getting something that the person won't like. [Sighs. Newcome. Any lady must like this brown, madam. Just feel the texture of this piece of goods, and take the trouble to examine the quality. Why, I have never in all my experience sold a piece of goods of such a class at a cent less than two dollars a yard—never. Maud. It is very fine. Ethel (vaguely eying the goods behind the counter on the shelves). Is that a piece of claret-colored that I see up there? Newcome (lays down the browns with a faint sigh of reluctance). Yes, oh, yes. Ethel. Never mind to get it down. Newcome. No trouble in the world to show anything; that's what I am here for. (Sighs as he attains the clarets and fetches them to the counter.) Rich shades; ten tints in these also, calculated to suit any taste. Maud. I always did like claret.
- 59. Ethel. Yes, it is so becoming. Maud. It has such a warm look, too! Ethel. Now, that—no, this one—no, please, that darker piece—yes. Maud, dear, that made up with plush and garnet buttons and buckles —Oh, did I tell you I saw some such lovely garnet trimmings at Blank's last week, only seventy-five cents a yard, just a perfect match for this. Wouldn't it be too lovely for anything? Maud. Indeed it would. I am almost tempted myself. Claret is my color, you know. Newcome. A splendid shade, madam, and only just two dress lengths left. Ethel. Is this the same goods as the others? Newcome. The very same; all-wool imported suitings, forty-four inches wide, reduced from two-fifty a yard to only one dollar and a half. Maud. Wouldn't that be just perfect with that white muff and boa of mine, dearest? Ethel. Too startling, love. Do you know, I think you made a mistake in getting that white set. Maud. Why? Ethel. Too striking. Maud. Do you think so? Ethel. Yes. Of course it's lovely for the theatre and opera. Maud. It's awfully becoming. Ethel (to Newcome). Now, do you really sell as much claret color as you do green or brown this season?
- 60. Newcome. Oh yes, madam; if anything, more. You see claret is one of the standards, becoming alike to young and old. Why, a child might wear this shade. Claret will always hold its own; there is a change in the blues and the greens and the browns, but the claret is always elegant, and very stylish. Maud. I think so too. Ethel (meditatively). I do wonder if she would like claret better than brown. Newcome. I can show you the browns again, ladies. Ethel. Oh, never mind. Newcome. No trouble in the world. (Holds up browns and clarets both.) Now you can judge of the two by contrast. Maud. Both lovely. Ethel. Which do you like best, love? Maud. My dear, I don't know. Newcome. You can't go amiss, madam, with either of those, I am sure. Any lady must like either of them. Ethel. Oh, dear! I wish people would get well and do their own shopping; it is so trying! Maud. Horrid! Newcome. An elegant piece of goods, madam; will wear like iron. Ethel. What would you do, dear? Maud. I really don't know what to say. When does she want to wear it? Ethel. Dinner and theatre. Maud. By gaslight, then?
- 61. Ethel. Yes, of course. Maud. Does the gaslight change the shade much? Newcome. Just a trifle, madam; it makes it richer. Maud. Darker? Newcome. Just a half a tone. Ethel. Then that must be considered. Oh, dear! [Sighs plaintively. Maud. Why not look at it by gaslight, love? Ethel. Oh, I hate to give so much trouble! Newcome. No trouble in the world, madam—a pleasure. I will gladly show you these goods by gaslight, for I am confident you will only admire them the more. Here, boy (calls boy, and hands him a pile of goods), take these to the gaslight-room. This way ladies, please. (They cross the aisle and enter the gaslight-room, preceded by the boy, who sets down the goods and retires.) There! look at that! Isn't that a rich, warm, beautiful color! [Displays clarets. Maud. Lovely! Ethel. Yes, lovely—but (dubiously) I am so afraid she won't like it. Maud. It is very perplexing. Ethel. Yes. Oh, how sweet those browns do look in this light! Don't they? Newcome. Ah, I just brought over the browns, madam, for I thought you might care to see them too. [Displays browns.
- 62. Maud. How they do light up! Don't they? Newcome. Newest tints, every one of them. Not been in stock over a few weeks, and those browns have sold like wildfire. Ethel. For my own part I always did like brown. Maud. Yes, so do I. Ethel. It's so ladylike. Maud. Yes, and it's a color that is suitable to almost any occasion. Ethel. Yes. Now that lightest piece would be just too sweet, wouldn't it, made up with that new Persian trimming? Maud. Exquisite! Say, do you know I priced some of that trimming the other day. Ethel. Did you? how much? Maud. Awfully expensive! Five dollars a yard. Ethel. How wide? Maud. Oh, not more than four inches. Ethel. It wouldn't take much, would it? Maud. That depends on where you put it. Ethel. Well, just on the bodice and sleeves and collar. Maud. About two yards and a half. Ethel. Fifteen dollars? Maud. Yes. Newcome. This brown trimmed in the manner you mention, ladies, would be very elegant.
- 63. Maud. Yes, so it would. I wish now that I had looked more particularly at the browns out by the daylight. Newcome. It is easy to look at them again, madam, I am sure. Here, boy, carry these goods back to the counter where you got them. (Boy crosses, laden with goods; Newcome and ladies follow.) That's it. (Boy retires.) Now, madam, just look at that shade by this light. Isn't that perfect? Ethel. Yes, it's lovely, but— Maud. Did she say she wished a brown especially, dear? Ethel. No, she left it to me entirely. Maud. How trying! Ethel. Yes. I—I really, you know. I don't dare to take the responsibility; would you? [Newcome's arms falter slightly in upholding the goods. Maud. Frankly, my love, I think shopping for anyone else is something dreadful. Ethel. It is so trying and so embarrassing. I don't dare really to get either (Newcome's arms fall helpless; he sighs) one of them. Maud. They are lovely, though; aren't they? Ethel. Yes, if (Newcome revives a little) I thought she would really be satisfied. [He essays once again to hold up the browns. Maud. But, dear, they never are. [His arms again droop. Ethel. No, never. No matter how much trouble you take, or what pains you are (he sighs feebly) at (he totters), they are so
- 64. ungrateful. Maud. Yes, always. Ethel. Well, I believe we can't venture to decide this morning (he staggers) about the shade. We will very likely return to-morrow. [He raises a weakly deprecating hand. Maud (aside, as the two ladies are going). Well, we got off quite nicely. Ethel. Yes, didn't we! I wouldn't be seen in either of those horrid things; would you? Maud. No. [Newcome falls to the earth with a groan of despair; the Chorus rush forward and gently raise him in their arms. As they bear him off, they sing, in a doleful and yet half- malicious fashion: Chorus. Poor Newcome! You are not the first man they have ended, And left on the cold ground extended; Or to whom they have sweetly pretended, On whose taste they have weakly depended;— Whom they've left on the cold ground extended, Minus money they never expended, On goods that they never intended To buy, Heigh-o, heigh, O—O—! [They retreat, C., as the ladies exeunt, R., L. Music pianissimo as curtain falls.
- 66. IRISH NORAH TO ENGLISH JOHN. (Her theory of Home Rule under the Union.) It manes, and shure and where's the harm? Said Nora to her spouse; It manes: if you must mind yer farm, That I shall mind me house.
- 67. BELLA'S BUREAU.[4] A STORY IN THREE SCARES. SCARE THE FIRST. I almost flung myself into Dick Vandeleur's arms when he entered my library that evening. Can you imagine why I sent for you in such a deuce of a hurry? I blurted out, embracing him effusively in my pleasure at seeing him. Well, I did think there might have been a woman in the case, he drawled, in his deliberate way, stopping to adjust his neck-tie, which had worked its way over his ear during the struggle. But then, as I happened to have acted as your best man only two months ago, when you married the most charming of women, why, b'Jove, I— Well, it is a woman, I groaned, cutting his speech short. The devil! Yes, and the very worst kind, I fancy, if thoroughly aroused. But, my deah boy, with such a wife it's—it's—it's— Yes, it's all that and a good deal more, I growled, gloomily. Don't add to my misery with your ill-timed reproaches. Richard, a back number of my unsavory career has turned up to deprive me of my appetite and blight my being. You remember Bella Bracebridge, of the nimble toes, at whose shrine I worshipped so long and so idiotically? Well, I received a letter from her only yesterday. No!—incredulously.
- 68. Yes. What!—little Bella who used to caper around in such airy garments at the Alhambra? The very same. I only wish I could be mistaken, with a despairing groan. It seems she married money and retired from the stage. By some means she disposed of her husband, and is now a rich and probably good-looking widow. She has purchased an estate within half a mile of here, and is going in heavy for style. She wants to make me the stepping-stone to social success; she sighs for the purple penetralia of the plutocracy. See what a predicament I am in! To introduce her in this house would plant the most unjust suspicions in Ethel's Vassarian mind, while her mother, Mrs. McGoozle, might institute awkward inquiries into the dear, dead past—with a shiver of anticipation. Now, my dear Vandeleur, that woman means mischief. She has got about a hundred of my letters breathing the most devoted love: if dear Ethel got a glimpse of a line she would go into hysterics. Bella has hinted, even politely threatened, that unless I show her some attention, which means introducing her to my wife's circle of friends, she will publish those letters to the world or send them to the dramatic papers. Now you must help me out of this scrape. Delighted to be of any service, I'm sure, tapping his boots impatiently with a jaunty little cane. But, really, you know, I don't see— Why, it's easy enough. Don't you remember we were once the pride of the school because we robbed watermelon patches so skilfully? What a narrow shave that was in the apple orchard the night before commencement, when you— Yes, yes, I remember, deah boy; but what have those childish pranks got to do with the present case? We don't want to rob an apple orchard—by way of mild protest.
- 69. It is another kind of fruit that we are after—the fruit of youthful follies. Here, opening a cupboard and throwing out two pairs of overalls somewhat the worse for paint, two jumpers ditto, and several muddy overshoes, Vandeleur, if you love me put these things on. I fancy I can see him now adjust his glass and survey me with bulging eyes. I certainly did have nerve to ask that famous clubman, so irreproachable in his dress, to assume such inartistic and plebeian garments. It took a great deal of palavering before I could persuade him that I was lost unless he consented. How he grunted as he reluctantly laid aside his silk-lined white kersey coat and evening dress, and tried to put on the overalls with one hand while he held his aristocratic aquiline nose with the other. Really, I hope I shan't be found dead in these togs, he remarked ruefully, as he surveyed himself in the glass. What would Flossy say? and how the chaps at the Argentine would wonder what I'd been up to! I cut short his speculations by thrusting a soft slouched hat on his head and dragging it down over his eyes. There now! I said, standing off and contemplating him critically and admiringly; you have no idea, my boy, how becoming this costume is. One might imagine you had been born a stevedore. He looked rather sour at this doubtful compliment, and hitching up his baggy trousers, asked, Well, what is the next misery? It is twelve o'clock, I said, referring to my watch. My wife has gone to bed. Like Claude Duval, we will take to the road. After a stiff libation of brandy and soda we stole softly downstairs and found ourselves in front of the house. Only one light glimmered in the black pile, where Ethel was going to bed.
- 70. Where away? asked Vandeleur, as I turned the path. To storm Bella's bureau, I cried, leading the way through the dark. SCARE THE SECOND. With much difficulty we found ourselves at last in the spacious grounds of Bella's estate. I had laid my plans carefully the day before, and there seemed no possibility that they would miscarry. By liberal fees I had learned from her butler that she was to spend that night in New York with a friend, and for a further consideration he offered to leave one of the drawing-room windows open so that we should have a clear field. Everything seemed to be working beautifully, and I already felt the coveted letters in my grasp. We found the French window ajar, and with tremulous hearts stepped over the sill and into the room. After several collisions with the furniture, of which there seemed to be what we thought an unnecessary amount, we finally scraped our way into the hall. Here was a quandary. We were in a hall, but what hall? Whether the stairs led in the right direction there was no one present to consult. We walked or rather crawled up them, nevertheless. I tried the first door on the landing, and was rewarded with Is that you? by a female voice that sent us scuttling along the passage in undignified haste. Well, at last, after many narrow escapes from breaking our necks, we reached Bella's room. I knew it the moment I saw the closet full of shoes. Bella was always proud of her feet, and had, I believe, a pair of boots for every hour of the day. To make things even more sure that I had arrived at the chaste temple of my former flame, there was the famous bureau of ebony inlaid with ivory—that bureau which contained enough of my inflammatory letters to reduce it to cinders.
- 71. Can you regard that bureau with equanimity? I exclaimed, unconsciously assuming a dramatic attitude. Does it not recall your vanished youth—the red horizon of your adolescence? Ah, I cried, overcome by the sight of that familiar bit of furniture, how often have I slid a piece of jewelry into that top drawer as a surprise for Bella! Her delighted shriek which followed the discovery rings in my ears even now. Oh, halcyon days of happy holiday, mine no more, can a lifetime with a funded houri wholly fill your place? That's all very well, cried Vandeleur, who can assume a disgustingly practical tone when he wants to. While you are rhapsodizing here over your poetical past, some stalwart menial may arrive with a blunderbuss, and fill our several and symmetrical persons with No. 2 buckshot. Perhaps Bella may have missed her train or her friend. She might return here at any moment and surprise us—looking around him uneasily. Anybody would think that you had never been in a boudoir at this time of night, I retort savagely. I begin to pull out the drawers of the bureau, breaking locks in the most reckless way, and tossing the contents of these dainty receptacles about in the most utter confusion. Vandeleur, with his eyeglass adjusted, is poking into everything in the closet as if he were looking for a mouse, only pausing now and then to glare around with an apprehensive shiver. Dear me, I soliloquize, while the contents of those bureau drawers are tossed here and there in the fever of my search. How everything here reminds me of the past! She has even preserved the menu card of that memorable dinner at Torloni's; and here—here is a lock of brown hair tied with a pink ribbon! I really believe it must be mine! My deah boy, howls Vandeleur, shaking me by the arm vigorously, will you cut short your soliloquy? Is this a time for poetry, when we might get ten years if we were found burglarizing this house?
- 72. I pay no attention. And here is the steel buckle from her shoe that fell off the night we danced together at the French ball. Poor dear Bella! that was not the only dance we led where folly played the fiddle!—with a thrill of reminiscence. If you don't find those letters in just two minutes, interrupts the dreadful Vandeleur, I shall post for home. In one second, my boy—one second. Now I examine the bureau carefully for a concealed drawer. I seem to have ransacked every corner of that precious article in vain. Visions of Bella's vengeance flash before my eyes. I can see the demoniac smile on her face as she gloats over my downfall. The white wraith conjured up by the thought of those fateful letters fills me with a mad fury, and I long to dash that hateful bureau into a thousand pieces and flee the house. But the demolition could not be executed noiselessly, and the situation is perilous enough already for a man of my delicately organized constitution, with a heart that runs down with a rumble like a Waterbury movement; so I think I won't break the bureau. I renew my mad search for the missing drawer, that seems to be of a most retiring disposition, as drawers go. I bethink me of stories of missing treasure: how the hero counted off twenty paces across the floor, and then dropped his dagger so that its blade would be imbedded in the wood, and then dug through several tons of masonry, until he found a casket, sometimes of steel, sometimes of iron, and sometimes of both. And then he did a lot more mathematical calculating, and pressed a knob, and there you are! Ah! a thought—I had forgotten to apply myself to the moulding of the bureau, as a hero of the middle ages would have done under the circumstances.
- 73. I begin from side to side, up, down, and around. Ha! ha! at last! A little drawer shoots out almost in my face, startling me like a jack-in- the-box. A faint perfume of crushed violets salutes my nostrils. The letters— they are there in the bottom of the drawer! I know them too well by the shape of the square large envelopes. They cost me many a dollar to send through the stage-door by the gouty Cerberus at the gate when Bella trod the boards. I reach out my hand to seize them, when an awful scream causes me to stagger back in dismay. Bella Bracebridge, in a jaunty travelling dress, stands in the doorway in the attitude of a tragic queen—her eyes flashing, her bosom heaving, just as she looked the day she asked for a raise in salary and didn't get it. She steps towards me: I retreat, transfixed by her defiant attitude. She fear a common burglar? Never! I know she intends to seize me and scream for help, and I am afraid, too, that she may recognize my face. So I step back—back, edging towards the window. She reaches out her hand to seize me, then totters and falls in a dead faint. I look around for Vandeleur. He has lost all presence of mind; is staring at the figure on the floor, with wild, dilated eyes, and an expression of hopeless idiocy on his face. I can hear people moving below stairs. Her scream must have aroused the house. Vandeleur, shaking him by the arm, we must run for it. Do you understand? Ten years! Hard labor!—the last words hissed excitedly in his ear. What? where? who? he mumbles, with a face as expressive as that of codfish.
- 74. I rush to the balcony to see if we can make the jump below. It is dark, but the leap must be made. Better a broken leg than a ball and chain on a healthy limb for years and years. I drag Vandeleur in a helpless condition out on the balcony, boost him up on the railing, and push him off. Then I leap after him. Fortunate fate! We fall into a clump of blackberry bushes, and not a moment too soon. Lights flash out from above. I hear the hum of excited voices, Bella's calm and distinct above the rest, as she gives the ominous order, Let those bloodhounds loose! Ugh! We scramble out of the bushes in the most undignified haste, leaving most of our outward resemblance to human beings on the thorny twigs. Then helter-skelter over the fields and hedges, stumbling, staggering, and traversing what I suppose to be miles of country. Vandeleur is snorting like a steam calliope in bad repair, and I am breathing with the jerky movement of an overworked accordion. I can go no farther, he exclaims, dropping down in a huddled heap at the foot of a scrubby pine-tree like a bag of old clothes. I don't feel much in a hurry either, but I try to infuse some life into him by hustling him and shaking him in a brutal and unsympathetic manner. Do you hear that? I howl in despair as the baying of the bloodhounds rolls towards us over the meadow like muttered thunder. There is nothing to do but climb this tree, unless you want to furnish a free lunch for those brutes. Free lunch? get me some, he mumbles, relapsing into his old idiotic state again. Then I fall upon that unfortunate man in a fury of rage, and pound him into a consciousness of his danger. He consents at last to be pushed or rather dragged up in a tree, whose lowest limb I straddle with a feeling of wild joy and ecstasy
- 75. just as the hounds rush past below, their flashing eyes looking to me just then as large as the headlights of a host of engines. Let's go home now, again murmurs the helpless creature at my side, shaking so on the limb that I am compelled to strap him there by his suspenders. Ain't we going home? he chatters. I want a good supper, and then a bed—bed, lingering on the last word with soothing emphasis. Oh, you'd like a nice supper, would you? I growl. Well, those bloodhounds are after the same thing. Perhaps you had better slide down the tree and interview them on the chances. Then one or the other of you would be satisfied. But they've gone away. Well, you needn't think you have been forgotten, just the same. Don't you see, wretched man, that the morning is breaking, pointing to the east, where the sun had begun paintin' 'er red. Once in the high road we should be discovered at once; here at least we are safe—uncomfortably safe, as I moved across the limb and impaled myself on a long two-inch splinter with spurs on it. He fell into a doze after that, only rousing himself now and then to utter strange croaking sounds that frightened me almost as much as the baying of the bloodhounds. I think I fell asleep too for a few moments, for when I was roused by an awful yell proceeding from my companion I found that he had burst his bonds and fallen out of the tree, while the bright sun was shining in my eyes. Visions of Ethel's face over our charming breakfast-table rose before me, and I seemed to scent afar off the steam of fragrant mocha in a dainty Sèvres cup as she held it towards me. The thought of that morning libation settled the business. I would march stalwartly home—yea, though a thousand bloodhounds with dangerous appetites barred my way!
- 76. I slid down the tree and found Vandeleur still asleep. I don't believe that even the fall had waked the poor fellow up. I had only to whisper the word Breakfast in his ears to have him start as if he had received a galvanic shock. Where? he asked, with tears in his eyes. Home. We crawled along through the bushes in the wildest haste our poor disjointed and almost dismembered bodies would carry us; like a pair of mud-turtles who had seen better days did we take to all- fours. Fortunately, my place was not far away, and we had just strength enough to crawl up on the porch and fall against the door heavily. Breakfast, I gasped, as Ethel's lovely face appeared suddenly at my side like a benignant angel's. What—what can I get you? murmured the dear girl, in an agony of mind, hurrying here and there, her eyes suffused with tears. Bloodhounds! murmured Vandeleur, relapsing into idiocy. SCARE THE THIRD. If you have ever had the fortune to be married to a Vassar graduate of the gushing and kittenish order, between nineteen and twenty, you will understand how difficult it was to explain my dilapidated appearance that memorable morning. The ingenuity of my fabrications would have stocked a popular romance writer with all the modern conveniences; and I am sure the recording angel must have had difficulty in keeping pace with my transgressions unless he or she understood short-hand.
- 77. Vandeleur took an early opportunity to escape to the city, knowing very well that he would be held accountable for my degraded and dilapidated condition. The friends of a married man always are held responsible by his wife for any of his moral lapses, no matter when or where they may occur. If I had only succeeded in my undertaking I might have viewed even my wounds—of which there were many—with some equanimity. But to have suffered in vain was enough to try the strongest soul; and I am afraid I was unnecessarily brusque to Ethel when she insisted on soaking me hourly in the most horrible liniments of her mother's decoction. I was pickled for about a week by her fair hands, and had become so impregnated with camphor and aromatic compounds that I exhaled spices like an Eastern mummy or a shopworn sachet-bag, and longed to get away from myself and the drugstore smell that clung to me closer than I ever want my brother to cling. I consented to the embalming process, because I wanted to look respectable when Ethel's mother, Mrs. McGoozle, put in an appearance. I knew I could not so easily satisfy her mind regarding that night of folly without the sworn affidavits of half-a-dozen reputable citizens. She said I wrote so much fiction that it had become a habit with me never to tell the truth. My eyes had just begun to lay aside mourning when I received at the dinner-table one stormy night the local paper. I took it for my wife, who had a penchant for reading the patent-medicine advertisements; but on the present occasion I displayed an unholy eagerness to get at its contents. More misery! More horrible complications! Almost the entire sheet was given up to a description of the burglary. There was a picture of Bella's house and of Bella herself; of the cook, of the coachman—yes, and even of the bloodhounds. I had puzzled my brain since that night in trying to imagine why the hounds had sped past our tree, our noble tree, instead of gathering in convention at its base and talking the matter over among
- 78. themselves while we starved to death upstairs. The paper gave the solution of the problem. They were pursuing the trail that led to our milkman's farm—the poor creature of whom I had basely borrowed our suits and overshoes. The worthy man had been arrested and haled before the nearest justice of the peace, and had he not been able to prove an alibi to the effect that he was watering his cows at the time, he would have been summarily dealt with. But he had held his peace about my share in the transaction—bless him! and being a thrifty man, had brought a suit against Bella for threatening his life with her dogs. Yet I had no cause for congratulation, for now I was in the milkman's power as well as Bella's; and the very next day the honest fellow put in an appearance, very humble and yet very decided, and insisted that I should present him with Ethel's prize Frisian cow as a premium on his silence. And I had to consent, though my wife had hysterics in parting with the animal, and sobbed out her determination to tell Mrs. McGoozle everything when that lady arrived in a few days. This may not sound very terrible to you, but I knew the dreadful import of her words. There was a flash of light through the gloom of my suicidal thoughts the next morning that made my heart beat high with hope. I read in the morning paper that Bella, the cause of all my trouble, was dead, and that there was to be a sale of her effects at a New York auction-room the next day. Of course that dreadful bureau was in the lot, and I knew that if it fell into unscrupulous hands there was enough material in that little drawer to stock a blackmailing establishment for years and years.
- 79. I took the first train for the city on the day of the sale. The bureau— Bella's bureau—was just being put up as I entered the place. I had a thousand dollars in my pocket, so I felt rather contented in mind. The bidding on the bureau began in a discouraging way. The hunger of the crowd had been appeased before I came, and they displayed a lukewarm interest in the bureau. I bid two hundred dollars finally to settle the argument. I was tired of the delay. I wanted to settle forever the incubus that preyed upon my spirits. Two hundred, I cried exultantly. Three hundred dollars, came in quiet tones from the corner of the room. The words seem to ripple in an icy stream down the back of my neck. Could it have been the echo of my voice that I heard? Four hundred, I cried uneasily. The terrible thought flashed over me, that perhaps another lover had turned up, who believed that his letters were in the bureau, and was just as anxious to get it as I. Horrible! Four hundred is bid for this beautiful Louis Fourteenth bureau, howled the auctioneer, repeating my bid. Why, gents, this is a shame: it's— Five hundred, said the voice from the corner, in calm, cold tones. Ah, if I could slip through the crowd and throttle his utterance forever. Six hundred, I screamed, in desperation. Then my unseen foe woke up and we began to bid in earnest. Six, seven, eight hundred, ran the bids. In one of the lulls of the storm, when the auctioneer began to wax loquacious regarding the beauties of that bureau, I slipped secretly around to the cashier's desk.
- 80. Would he take a check? I implored. No, he would not; and I thought he wore a triumphant glitter in his fishy eyes. The terms of the sale were cash: it was to conclude that day. I turned away, sick at heart. A thousand! I cried, in desperation, staking my last dollar. There was a moment's ominous silence. I began to feel encouraged. I watched the fateful gavel poised in the air, with my heart in my teeth. It wavered a moment, then began to slowly descend. Never had I seen such a graceful gesture defined by man as the freckled fist of the auctioneer described at that moment of hope. Twelve hundred, croaked the demon in the corner. The crowd blended into a pulp of color. I fainted. I lingered about the city all that night, searching in vain for a lethean draught at the haunts where consolation is retailed at two hundred per cent profit. I did not find the nepenthe I sought for anywhere on draught, so I went home in disgust. Ethel received me in her usually effusive manner. She knows I object to being hugged at all hours of the day, yet I have never been able to cure her of that affection-garroting process so much in vogue with young wives of the gushing order. What do you think? she chirped, when I had staggered to a chair in a half-strangled condition. Dear mother has just sent us the most beautiful present— Oh, I suppose so, I sneer savagely. She generally does present us with something beautifully useless. Perhaps this time it's a dancing- bear, or a tame codfish—with a wild laugh. Oh, how can you talk so! lifting a dab of cambric to her nose with a preliminary sniff that is generally the signal of tears, according to our matrimonial barometer. You know dear mother is so fond of you.
- 81. Well, it's a case of misplaced affection, I growl, lounging out of the room just in time to avoid the rising storm. I dash upstairs and smoke a cigar in my own room. Then I feel better, and stroll into Ethel's boudoir, resolved to pitch her mother's present in the fire if it doesn't suit me. She ought to be suppressed in this particular. Wha—what! No—yes, it is! The bureau, Bella's bureau, stands in the chaste confines of Ethel's satin-lined nest. I fling myself upon it, tear the little drawer open—hurl the bundle of letters into the grate with a cackling laugh. Ethel enters timidly just then, and looks first at me and then at the burning papers with doubt and wonderment in her blue eyes. I have been paying some old debts, I say, with an uneasy laugh. These are some of the I.O.U.'s you see burning. She lays a soft little arm around my neck and a curly head on my immaculate shirt-front. Oh, spotless mask for such a darksome heart! I wonder she cannot catch the sound of its wicked beating. I have been worried about you lately, dear, she whispers, with a tender tremor in her voice. I thought perhaps you might—you might—have become entangled with some other—other— Then she burst into tears. How often must I tell you, darling, patting her cheek softly, that you are the only woman I ever loved? Oh, Jack! Ernest De Lancey Pierson.
- 82. A SHOT ON THE MOUNTAIN.
- 83. An eagle drifting to the skies To gild her wing in sunset dies, To float into the golden, To swing and sway in broad-winged might, To toss and heel in free-born right, High o'er the gray crags olden. A dark bird reaching on aloft, Till far adown her rugged croft Lies limned in misty tracing— Till, riding on in easy pride, Her cloud-wet wings are ruby pied, Are meshed in amber lacing. An eagle dropping to her cave On dizzy wing through riven air, A bolt from heaven slanted; A startled mother, arrow-winged, A mountain copestone, vapor-ringed, An eyry danger-haunted. An eagle slanting from the skies To stain her breast in crimson dyes Beneath the gilt and golden; A shred of smoke—the gray lead's might— A folded wing—the dead bird's right— Abreast the gray crags olden. The blush light fades along the west, The night mist rolls to crag, to crest, To cowl the ghostly mountain; Black shadows hush the eyry's calls; Below, a broad brown pinion falls— The last light from the fountain. J. W. Rumple.
- 85. EDITORIAL DEPARTMENT. PURIFYING THE POLLS BY LAW. The edifying efforts made by Congress to throw guards about the ballot would be encouraging were they based on a little knowledge of the fact, and the reason for it. As it is, the be-it-enacted agreed on is little better than a solemn protest. Our learned law-makers would enjoy greater progress if they would remember that we have had for a century all the law necessary to punish such corruption, and that the trouble lies in our inability to enforce its provisions. What is really wanted is a tribunal to try and enforce the stringent enactments already in existence. This does not now exist. When a candidate for Congress corruptly purchases enough votes to secure his return to either House, he knows that such Chamber, being the judge of such applicant's qualifications, forms a court without a judge to give the law or an impartial jury to render a verdict. The Committee on Elections in either House is made up of the Democratic or Republican party, and so the jury is packed in advance. This is not, however, the only evil feature in the business. There is probably no organized body so ill-fitted for adjudication upon any subject as Congress. Returned to place by parties, the members are necessarily partisans. Their tenure of office is so brief that they have no time in which to learn their legitimate duties through experience, and these duties are so numerous, to say nothing of being encroached upon by services entirely foreign to their positions, that they have no opportunities for study. The consideration then of any subject from a judicial point of view is simply impossible. It is touch and go with them, and the touch is feeble, and the go hurried. It
- 86. seems that a case of purely judicial sort has no place in Congress; and yet we have seen an instance—for example, in the New Idria contention—where the courts had been exhausted, from an Alcalde to the Supreme Court of the United States, and yet the complainant, worsted in every one of these tribunals, came through the lobby into Congress, and for over ten years kept that body in a tumult. Of course this was kept alive by the corrupt use of stock to the extent of ten millions, based on the credit of a company that would be such when Congress gave its illegal approval. This fact alone proves the dangerous and uncertain character of a legislative body that takes on judicial functions. When a contested election goes before the standing committee called into existence as a court, it passes into a secret committee- room, where the so-called evidence, put on paper, is supposed to be considered. What would be said of a jury impanelled avowedly from the party of one side, and then made into a court to sit and deliberate with closed doors against the public? It is true that the finding is shaped into a report and goes before the House. But no member of that body, especially of the House of Representatives, has either time or opportunity to read the evidence, or even to listen to the arguments made by contestants upon the floor. That tribunal has lost all power in its loss of public confidence. It not only brings the law into contempt, but itself into such disrepute that its findings are worthless. This is the condition of Congress in public opinion. So far as contested elections are concerned, it is regarded with contempt. To make matters worse, and pay a premium on vice, the losing party is allowed the same mileage and pay given his successful competitors. If all contests were turned over to the United States courts, to be tried in the locality where the wrong complained of was done and the witnesses live, there would be few contested elections, and some chance given to punish bribery and other corruption.
- 87. Again, the prohibition against the subscription or payment of money has exceptions that open wide the doors to corruption. To say that money may be used for any purpose is to leave the evil precisely where the law-makers found it. It were better to have the government furnish the tickets, as the government supplies the ballot-boxes, rents the polling-places, and pays the officials for their services. The ballot is as much a necessity to the machinery of election as the boxes; and because it would be difficult and troublesome to supply them is far from saying that it is impossible. Then, to punish both bribe-giver and bribe-taker in the same way is to throw a guard about the iniquitous transaction. The bribe-taker should go acquit. Of course this would be in a measure opening a door to blackmailers, and make the candidacy extremely dangerous. Such it ought to be. The sooner we put a check on the shameless solicitations for office the better it will be for the Republic. Let the offices, as of old, in the purer days of the fathers, seek the man— and not the man, as now, the offices. If the effect of this would be to drive timid, decent men from office, it would not be worse than the present system. A candidate for the House of Representatives must not only pay his two years' salary in advance to heelers, as they are called, but must get drunk in every saloon in his district. We cannot make matters worse, and there is a chance in a change for an improvement. True reform to be effectual must be radical. A compromise with evil is a surrender to hell. To cut a poisoned shrub even to the ground relieves the eye for a time, but the root is made more vigorous by the trimming. The constitutional governments of Europe have rid themselves of bribery and other corruption by digging out the roots. This is the only course open to us. When members of the House can bribe their way to place, when Senate chairs are sold in open court, when it calls for only two millions to purchase the Presidency, and all done by men of high social position, we have reached the lowest level, and our great Republic is a mere sham and a delusion. We are
- 88. not menaced with the loss of liberty and guaranteed rights. They are gone. THE MUGWUMP ELEMENT. The purchase of the Presidency in open market, now generally recognized, is less disheartening than the apathetic indifference in which such corruption is regarded by the people. In all communities men may be found to buy, and men to sell, the sacred privilege upon which our great Republic rests; but it is rare—so rare that this experience is almost without precedent—that good citizens, knowing the nature of their free institutions, are willing to have them destroyed without an effort in behalf of their preservation. To get at not only the fact but the reason of it we must remember that politics to the average citizen has all the fanaticism of religion, and all the fascination of gambling. We have the country divided in two hostile camps, and in these organizations themselves we have lost the objects for which they were organized. This is the tendency of poor human nature the world over. It is probably more pronounced in religion than in any other form. A man will not only fight to the bitter end, but die as a martyr, for a sect whose dogmas he has never read, or, if read, fails to comprehend. Politics is our popular religion. Taking the great mass of our citizens, we are pained to write that it is about our only religion. We say that we have two hostile camps, in both of which the objects for which they were organized have been entirely lost. The ordinary Republican can give no reason for being such save that he is not a Democrat, and the Democrat has the same reason, if it may be called such. Each will avow, without hesitation, that the other camp is made up of knaves and fools. The folly of thus designating over half our entire voting population does not strike the partisan. Parties, however, are not called into existence and held together through intellectual processes. They are founded on feeling. For
- 89. years and years the brightest minds and purest characters preached, with burning eloquence, upon the wrongs of negro slavery, and got ugly epithets and foul missiles in return, if indeed they were listened to at all. At a moment of wild frenzy an armed mob at Charleston shot down our flag. In an instant the entire people of the North rose to arms, and a frightful war was inaugurated. The flag sentiment outweighed the Abolition arguments. It is not our purpose to give the philosophical view of that contest. We use it only, so far, in illustration. The sectional feeling that brought on that armed contest continues in another direction, and divides the two great parties. It is so intense that each is willing to see the republic under which we live utterly destroyed, so that one may be conquered or the other defeated. We are all agreed that the ballot is the foundation-stone of our entire political structure. On this was built the form of government given us by the fathers, and was the grand result of all the blood and treasure, of life and property, so patriotically poured out in the Revolution that made us independent. Yet this ballot is openly assailed, its processes corrupted with money, and its usefulness entirely destroyed, without arousing the indignation of an outraged public. Men of wealth, of high social position, members of churches, and leaders in what are called the better classes, subscribe and pay the money knowingly that is to be used in the purchase of floaters in blocks of five or more, while voters, well-to-do farmers, and so- called honest laborers are organized willingly into blocks and shamelessly sell that upon which they and their children depend for life, liberty, and a right to a recompense for toil. When the result is announced bonfires are burned, and loud shouts go up amid the roar of artillery, expressive of the joy felt in such a triumph. These men of means—it makes no odds how the means are accumulated—are not aware that in this they are cutting away the foundations under their feet, and that, too, with ropes about their necks. Their only security, not only to the enjoyment of their property but to their lives, lies in the very government they are so
- 90. eager to destroy. We have called attention to the fact that humanity suffers more from an inequality of property than from an inequality of political rights. These last are rapidly getting to be recognized and secured in constitutions throughout the civilized world. Kings and emperors have come to be mere figure-heads above constitutions, and the political dignity of the poor man is generally acknowledged. But the poor man remains, and the castle yet rears its lofty front above the hovels of the suffering laborers. Humanity is yet divided between the many who produce all and enjoy nothing, and the few who produce nothing and enjoy all. This is the inequality of property, and governments yet hold the sufferers to their hard condition. It is called law and order, as sacred in the eyes of the Church as it is potent in courts of justice. There is no government so poorly fitted to the execution of the hard task of holding labor down as this of the United States. In Europe through the dreary ages the masses have been born and bred to their wretched condition. With us, on the contrary, there has been a great expenditure of toil and treasure to teach labor its rights. In Europe great armies are organized and kept upon a war footing for police duty. We have no such conservative force upon which to rely in our hour of peril, and yet so far our government has held sway through our habitual respect for that which we created. These wealthy corruptors are rapidly destroying this respect. They are teaching the people that their ballots are merchantable products, and their ballot-box a rotten affair. Violence follows fraud as surely as night follows day, or a thunderstorm a poisoned atmosphere. The day is not distant when these millionaires will be hunting holes in which to hide from the very mobs they are now so assiduously calling into existence. God in his divine mercy forgives us our sins when we are repentant, but the law that governs our being—called nature—knows no forgiveness. The wound given the sapling by the woodman's axe is barked over, but that cut, slight as it seems, remains, and may hasten decay a
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