![Other material in Chapter II includes investigations of paradifferential operators
on the new function spaces
Chapter III gives a sample of applications of some of the results of Chapters
I—TI to topics in PDE. We treat some linear PDE with rough coefficients, includ-
ing some natural differential operators arising on Riemannian manifolds with non-
smooth metric tensors. We consider the method of layer potentials on domains
that are not smooth (though not so rough as those considered in Chapter IV). We
also treat a couple of topics in nonlinear PDE, including inviscid, incompressible
fluid flow on rough planar domains and wave equations with quadratic nonlinear-
ities. We also discuss various div-curl estimates, including a number of estimates
of [CLMS]. Some of the work in this section, especially variable-coefficient results,
grew out of correspondence with P. Auscher, following up on our work in [ATI.
Other topics studied in Chapter III include the construction of harmonic coordi-
nates on Riemannian manifolds with limited smoothness, regularity results for the
metric tensor of a Riemannian manifold when one has estimates on the Ricci tensor,
and propagation of singularities for PDE whose coefficients are more singular than
C", but which still have well defined null bicharacteristics by virtue of Osgood's
theorem.
Chapter IV deals with the method of layer potentials on Lipschitz domains.
We establish the fundamental estimates of Cauchy integrals on Lipschitz curves of
[Ca2] and [CMM] (via a method of [CJS]) and extensions to higher dimensions
from [CDM]. We then discuss the Dirichlet problem for Laplace equations and
variants on Lipschitz domains. We consider operators with variable coefficients,
hence Lipschitz domains in Riemannian manifolds. Our treatment of this follows
[MT], though here we restrict attention to the simpler case of smooth coefficients,
whereas [MT] treats cases arising from C' metric tensors. This extends earlier
work of [Ve] and others on the flat Laplacian on Lipschitz domains in Euclidean
space.
Prerequisites for this work include an acquaintance with basic results on pseu-
dodifferential operators and some methods from harmonic analysis, including the
Littlewood-Paley theory. Sufficient material on these prerequisites could be ob-
tained from either [T2] or Chapters 7 and 13 of [T51. Indeed, this present work
can be viewed as a companion to 1T2].
Michael Taylor](https://image.slidesharecdn.com/21772-250430225601-a27b1f50/85/Tools-for-PDE-Pseudodifferential-Operators-Paradifferential-Operators-and-Layer-Potentials-Michael-E-Taylor-16-320.jpg)
![CHAPTER 1
Pseudodifferential Operators
with Mildly Regular Symbols
Introduction
Studies of pseudodifferential operators whose symbols p(x, satisfy a Holder
condition in x have been found to be very useful in PDE. A number of their prop-
erties and applications are investigated in [Bon], [Bour], [KN], [Meyl], [T2], and
other places. There has also been an interest in symbols whose i-dependence is
described by some other modulus of continuity; studies of this are made in [CM],
[Ma2], and [Ma3], for example. Here we study related problems. We concentrate
on measures of regularity just a bit better than mere continuity, with a secondary
interest in measures of regularity either barely better or barely worse than Lipschitz.
We measure such regularity in several different ways.
To be explicit, we consider four types of function spaces, associated with a
modulus of continuity w. First,
(0.1)
Second, with A(j) = w(22), we say
(0.2) E
where is a Littlewood-Paley partition of unity and 14'j (D) the associated
Fourier multiplier. Third, we say
(0.3) u E IN' — <Cw(2
where 4,,(D), which is an approximate identity as j —. oo. Fourth,
we say
(0.4) e <C21w(23).
When w(h) = h' (so A(j) = and r (0,1), then all the spaces (0.l)—(0.4)
coincide, due to well known results of Bernstein, Jackson, and Zygmund. For more
general r E R, (0.2) with A(j) = defines the Zygmund space C.
For other interesting w, such as
(0.5) w(h)
=
(for h E (0, 1/2]) with s > 0, or
(0.6) w(h)](https://image.slidesharecdn.com/21772-250430225601-a27b1f50/85/Tools-for-PDE-Pseudodifferential-Operators-Paradifferential-Operators-and-Layer-Potentials-Michael-E-Taylor-17-320.jpg)
![not all of these spaces coincide. In §1 we explore various properties of spaces of the
form (0.1)—(0.4), including containment relations among them.
In §2 we establish U-operator norm estimates for pseudodifferential operators
(0.7) p(x,D)u=
whose symbols p(x, 4) are of order 0 and satisfy various regularity conditions in x.
We consider symbol classes of the form XSrO, where
(0.8) p(x,4) e xsç"0
and X is some function space, such as etc. We take (4) — (1 +
in all cases except when (4) occurs as an argument of the logarithm; then we take
(4) = (4 + We also define in (2.52) symbol classes generalizing
the Hörznander classes Sfl. We recall that
(0.9) p(x,4) e sro
We produce results in terms of and compare these with results of
[CMI and [Ma3], given in terms of Neither set of results contains
the other. We can slightly extend the scope of the latter class of estimates by
recasting them in terms of
In §3 we decompose a symbol, such as in (0.8), into a sum of two terms. One,
p# (x, 4), has better smoothness properties, of use in further results on operator
calculus. The remainder, p5(x, 4), has no better smoothness, but does have lower
order (by a degree depending on the smoothness of p(x, 4)). Such symbol smooth-
ings follow a number of earlier models, including [KN], [Bon], [Mey2], and [T2].
A particular case of fundamental importance gives rise to Bony's paraproduct.
Constructions in §3 give rise to another class of symbol spaces, X defined in
(3.18)—(3. 19).
in §4 we analyze operators p(x, D) on variants of Sobolev spaces, of the form
(0.10) = {f A(D)f E L"(lft't)},
where A(4) = (4)cL(t) is a symbol satisfying a certain "ellipticity" hypothesis.
in §5 we treat operators p(x,D) on the spaces These are the spaces
from the list (0. 1)—(0.4) that seem best suited for analysis involving pseudodiffer-
ential operators. Largely for this reason, we have emphasized symbol spaces of the
form c0')sr0 and over other symbol spaces. Indeed, the class of spaces
C(A) is important even for analysis involving more usual sorts of pseudodifferential
operators. We mention that, for slowly varying 0,
(0.11) P e 0 S c 1 P:
which is a consequence of Proposition 5.7.
In §6 we study products of operators, particularly of the form p7 (x, D) where
p7(x, 4) arises from p(x, 4) E XSr(k) via symbol smoothing. We apply this to
commutator estimates in §7.
in §8 we discuss results about operators whose symbols p(x, 4) have Sobolev
space regularity in x.](https://image.slidesharecdn.com/21772-250430225601-a27b1f50/85/Tools-for-PDE-Pseudodifferential-Operators-Paradifferential-Operators-and-Layer-Potentials-Michael-E-Taylor-18-320.jpg)
![I.
In §9 we study operators arising from double symbols:
(0.12) Au(x) =
where a(x, y, has limited regularity in x and y. For example, parallel to (0.8),
we say
(0.13)
a Banach space of functions on 1W' x 1W'. There are similar definitions
of symbol classes X8r6, XSr(K)1 etc., for double symbols. While, as is well known,
if E 8r6,o E [0,1), then A in (0.12) can be written in the form (0.7),
with p(x, C such a reduction need not hold for symbol classes with limited
regularity, so a separate study of operators of the form (0.12) is called for. One way
operators of this sort arise is in the use of layer potential techniques for eliiptic PDE
on domains whose boundaries have limited regularity (a little better than C'). We
will explore this in §4 of Chapter III; this can be contrasted with the more difficult
study of such PDE on Lipschitz domains, taken up in Chapter TV.
In §10 we establish a commutator estimate of [CRW]. The approach we take
uses paraproducts, and follows a proof given in [AT]. This commutator estimate
plays an important role in div-curl lemmas, a topic that will be discussed in Chapter
III. It also is a key tool in the analysis of operators with vmo coefficients, the subject
of §11 of this chapter!
In § 12 we investigate operators on the Besov spaces (1W'), defined by
(0.14) 1 E B,,(1W') <cc.
with as in (0.2). In particular, the space (1W') is of interest because it
is contained in C (1W') but contains all spaces C(A)(R72) for which
Tn §13 study a rather general class of operators with coefficients in a function
algebra. One primary application of these results will be to cases involving Besov
spaces that are algebras.
In §14 we establish estimates on Pu, when P is a pseudodifferential operator
and u is barely more regular than continuous, of a sort pioneered in [BKM]. We
also produce improvements, involving estimates in and mention how such
results relate to constructions in Chapter 5 of [T2].
In §15 we present some generalizations of an interesting inequality that arises
in the study of CR manifolds in [Tu]. We interpret the estimates as commutator
estimates.
Tn §16 we look at some variants of Morrey spaces. We extend Morrey's imbed-
ding theorem to a class of Morrey-type spaces. We discuss advantages of expressing
these results in terms of spaces rather than the spaces more commonly
used for such imbedding theorems. We also study the action of various pseudodif-
ferential operators on these Morrey-type spaces.
1. Spaces of continuous functions
Suppose w is a modulus of continuity, i.e., a continuous, increasing function
on [0,00) such that w(0) = 0, and having the property that w does not vary too](https://image.slidesharecdn.com/21772-250430225601-a27b1f50/85/Tools-for-PDE-Pseudodifferential-Operators-Paradifferential-Operators-and-Layer-Potentials-Michael-E-Taylor-19-320.jpg)
![rapidly, i.e., that w(2h) Cw(h), V h (0, 1J. We denote by CW the space of
bounded, continuous functions on satisfying
(1.1) u(x + y) - u(x)J 5; Cw(PyI).
Next, if A(j) is a positive decreasing sequence, we want to define To do this,
we use a Littlewood-Paley partition of unity. Pick C°° (RTh), so that Wo(E) = 1
for CI 5; 1, 0 for CI 2. Set
(1.2) = = Wk(C) —
for k 1. Also set øo(C) = 4'o(C), so
(1.3)
Now we say u E C(A) if and only if
(1.4)
5; CA(j).
We also make the standing assumption that is slowly varying, i.e., A(j)
CA(j+1).
PROPOSITION 1.1. If w is a modulus of continuity, then
(1.5) A(j) = cw c
PROOF. Assume u C Cw. Then, for 1, since = 0,
= [u(x - y)- u(x)]
dy
5;
since is concentrated on yj $ (A more elaborate version of this argu-
ment is given in (1.42)—(1.43) below.)
Towards a converse, we have the following.
PROpOSITION 1.2. If AU) is a positive decreasing sequence such that E Mi) <
then
(1.7) c(A) c cc,
where
(1.8) cr(h) = inf
A(t) = 11(t) = >IMi).
j'çe](https://image.slidesharecdn.com/21772-250430225601-a27b1f50/85/Tools-for-PDE-Pseudodifferential-Operators-Paradifferential-Operators-and-Layer-Potentials-Michael-E-Taylor-20-320.jpg)
![this volume, Mr. Hulbert’s symposium includes “Government
co-operation in object-lesson road work,” by Martin Dodge;
Maurice O. Eldridge’s “Good roads for farmers,” Prof. Logan
Waller Page’s “The selection of materials for macadam roads,”
and E. G. Harrison’s “Stone roads in New Jersey.” There will be
a final volume devoted to an index.
+
—
Am. Hist. R. 10: 928. Jl. ‘05. 110w.
“The later volumes of the series present both the merits and
defects of the earlier ones. They are entertaining and often
suggestive, but always incomplete. The material is ill arranged,
and a surprising amount of it is reprinted from other books.”
+
—
Dial. 38: 322. My. 1, ‘05. 290w. (Review of v.
15 and 16.)
+
+
Engin. N. 53: 183. F. 16, ‘05. 300w.
N. Y. Times. 10: 132. Mr. 4, ‘05. 390w.
“A model of what an index should be.”
+
+
+
N. Y. Times. 10: 514. Ag. 5, ‘05. 140w.
(Review of v. 16.)
+
+
Outlook. 79: 449. F. 18, ‘05. 120w.
“[The index] is model work of its kind.”
+
+
Pub. Opin. 38: 838. My. 27, ‘05. 120w.
(Review of v. 15 and 16.)](https://image.slidesharecdn.com/21772-250430225601-a27b1f50/85/Tools-for-PDE-Pseudodifferential-Operators-Paradifferential-Operators-and-Layer-Potentials-Michael-E-Taylor-42-320.jpg)
Tools for PDE Pseudodifferential Operators Paradifferential Operators and Layer Potentials Michael E. Taylor
- 1. Tools for PDE Pseudodifferential Operators Paradifferential Operators and Layer Potentials Michael E. Taylor download https://ebookultra.com/download/tools-for-pde-pseudodifferential- operators-paradifferential-operators-and-layer-potentials- michael-e-taylor/ Explore and download more ebooks or textbooks at ebookultra.com
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- 5. Tools for PDE Pseudodifferential Operators Paradifferential Operators and Layer Potentials Michael E. Taylor Digital Instant Download Author(s): Michael E. Taylor ISBN(s): 9780821843789, 0821843788 Edition: Reprint File Details: PDF, 6.39 MB Year: 2007 Language: english
- 6. Tools for PDE Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials Michael ETaylor American Mathematkal Society
- 8. Mathematical Surveys and Monographs Volume 81 Tools for PDE Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials Michael E.Taylor American Mathematical Society
- 10. Editorial Board Georgia Benkart Michael Loss Peter Landweber Tudor Ratiu, Chair 1991 Mathematics Subject Classification. Primary 35S05, 35550, 42B20. The author was supported in part by NSF Grant #9877077. ABSTRACT. This book develops three related tools that are useful in the analysis of partial differ- ential equations, arising from the classical study of singular integral operators: pseudodifferential operators, paradifferential operators, and layer potentials. A theme running throughout the work is the treatment of PDE in the presence of relatively little regularity. In the first chapter we study classes of pseudodifferential operators whose symbols have a limited degree of regularity. In the second chapter we show how paradifferential operators yield sharp estimates on various nonlinear operators on function spaces. In Chapter 3 we apply this material to an assortment of results in PDE, including regularity results for elliptic PDE with rough coefficients, planar fluid flows on rough domains, estimates on Riemannian manifolds given weak bounds on the Ricci tensor, div-curl estimates, and results on propagation of singularities for wave equations with rough coefficients. Chapter 4 studies the method of layer potentials on Lipschitz domains, concentrating on applications to boundary problems for elliptic PDE with variable coefficients. Library of Congress Cataloging-in-Publication Data Taylor, Michael Eugene, 1946 Tools for PDE : pseudodifferential operators, paradifferential operators, and layer potentials / Michael E. Taylor. p. - (Mathematical surveys and monographs, ISSN 0076-5376; v. 81) Includes bibliographical references and index. ISBN 0-8218-2633-6 (alk. paper) 1. Differential equations, Partial. I. Title. 11. Mathematical surveys and monographs; no. 81. QA377.T37 2000 515'.353 — dc2l 00-036248 Copying and reprinting. Individual readers of this publication, and nonprofit libraries wting for them, are permitted to make fair use of the material, such as to copy a chapter for use .n teaching or research. Permission is granted to quote brief passages from this publication in provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication s permitted only under license from the American Mathematical Society. Requests for such )ermission should be addressed to the Assistant to the Publisher, American Mathematical Society, '.0. Box 6248, Providence, Rhode Island 02940-6248. Requests can also be made by e-mail to org. © 2000 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed io the United States of America. ® The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at URL: http://www.ams.org/ 10987654321 050403020100
- 13. Contents Preface ix Chapter 1. Pseudodifferential Operators with Mildly Regular Symbols I § 1. Spaces of continuous functions 3 §2. Operator estimates on If', 151, and bmo 17 §3. Symbol classes and symbol smoothing 31 §4. Operator estimates on Sobolev-Iike spaces 37 §5. Operator estimates on spaces §6. Products 54 §7. Commutator estimates 58 §8. Operators with Sobolev coefficients 61 §9. Operators with double symbols 63 §10. The CRW commutator estimate 75 §11. Operators with vmo coefficients 78 § 12. Estimates on a class of Besov spaces 82 §13. Operators with coefficients in a function algebra 86 §14. Some 8KM-type estimates 88 §15. Variations on an estimate of Turnanov 92 §16. Estimates on Morrey-type spaces 94 Chapter 2. Paradifferential Operators and Nonlinear Estimates 101 §1. A product estimate 105 §2. A commutator estimate 106 §3. Some handy estimates involving maximal functions 108 §4. A composition estimate 110 §5. More general composition estimate 112 §6. Continuityofu—*f(u)onH1'P 113 §7. Estimates on F(u) — F(v) 116 §8. A pseudodifferential operator estimate 118 §9. Paradifferential operators on the spaces 120 §A. Paracomposition 125 §B. Alinhac's lemma 132 Chapter 3. Applications to PDE 135 §1. Interior elliptic regularity 137 §2. Some natural first-order operators 148 §3. Estimates for the Dirichlet problem 155 §4. Layer potentials on C1" surfaces 159 Vt'
- 14. §5. Parametrix estimates and trace asymptotics 173 §6. Euler flows on rough planar domains 178 §7. Persistence of solutions to semilinear wave equations 183 §8. Div-curl estimates 186 §9. Harmonic coordinates 194 §10. Riemannian manifolds with bounded Ricci tensor 202 §11. Propagation of singularities 205 Chapter 4. Layer Potentials on Lipschitz Surfaces 217 §1. Cauchy kernels on Lipschitz curves 218 §2. The method of rotations and extensions to higher dimensions 228 §3. The variable-coefficient case 230 §4. Boundary integral operators 235 §5. The Dirichlet problem on Lipschitz domains 241 §A. The Koebe-Bieberbach distortion theorem 246 Bibliography 249 List of Symbols 255 Index 257
- 15. Preface Since the early part of the twentieth century, with the work of Fredhoim, filbert, Riesz, et al., the use of singular integral operators has developed into a range of tools for the study of partial differential equations. This includes the use of single and double layer potentials on planar curves to treat classical boundary prob- lems for the Laplace operator on a planar region and higher-dimensional extensions. It also includes the construction of parametrices for elliptic PDE with variable co- efficients. Fourier integral representations of these operators have provided many useful insights, though this method has not entirely supplanted the singular integral representation. When the use of the Fourier integral representation is emphasized, the operators are often referred to as pseudodifferential operators. Paradifferential operators form a singular class of pseudodifferential operators, particularly suited for applications to nonlinear PDE. Treatments of pseudodifferential operators most frequently concentrate on op- erators with smooth coefficients, but there has been a good bit of work on operators with symbols of minimal smoothness, with applications to diverse problems in PDE, from nonlinear problems to problems in nonsmooth domains. In this monograph we discuss a number of facets of the operator calculi that have arisen from the study of pseudodifferential operators, paradifferential operators, and layer potentials, with particular attention to the study of nonsmooth structures. in Chapter 1 we study pseudodifferential operators whose symbols have a lim- ited degree of regularity. We consider various cases, including measures of regularity just barely better (or just barely worse) than merely continuous, measures either a little better or a little worse than Lipschitz, and others. Function spaces used to describe the degree of regularity of symbols include Here C"' consists of functions with modulus of continuity w. The space with A(j) = w(23), is defined in terms of estimates on a Littlewood-Paley decomposition of a function. These spaces coincide for Hölder-Zygmund classes of functions, but they diverge in other cases. The space vmo is the space of functions of vanishing mean oscillation, and are certain Besov spaces. The interplay between some of these function spaces is itself a significant object of study in this chapter. The class of paradifferential operators, introduced in [Boni, has had a substan- tial impact on nonlinear analysis. In Chapter II we make use of paradifferential operator calculus to establish various nonlinear estimates, some of which have previ- ously been established from other points of view. My interest in organizing some of this material, particularly in was stimulated by correspondence with T. Kato. ix
- 16. Other material in Chapter II includes investigations of paradifferential operators on the new function spaces Chapter III gives a sample of applications of some of the results of Chapters I—TI to topics in PDE. We treat some linear PDE with rough coefficients, includ- ing some natural differential operators arising on Riemannian manifolds with non- smooth metric tensors. We consider the method of layer potentials on domains that are not smooth (though not so rough as those considered in Chapter IV). We also treat a couple of topics in nonlinear PDE, including inviscid, incompressible fluid flow on rough planar domains and wave equations with quadratic nonlinear- ities. We also discuss various div-curl estimates, including a number of estimates of [CLMS]. Some of the work in this section, especially variable-coefficient results, grew out of correspondence with P. Auscher, following up on our work in [ATI. Other topics studied in Chapter III include the construction of harmonic coordi- nates on Riemannian manifolds with limited smoothness, regularity results for the metric tensor of a Riemannian manifold when one has estimates on the Ricci tensor, and propagation of singularities for PDE whose coefficients are more singular than C", but which still have well defined null bicharacteristics by virtue of Osgood's theorem. Chapter IV deals with the method of layer potentials on Lipschitz domains. We establish the fundamental estimates of Cauchy integrals on Lipschitz curves of [Ca2] and [CMM] (via a method of [CJS]) and extensions to higher dimensions from [CDM]. We then discuss the Dirichlet problem for Laplace equations and variants on Lipschitz domains. We consider operators with variable coefficients, hence Lipschitz domains in Riemannian manifolds. Our treatment of this follows [MT], though here we restrict attention to the simpler case of smooth coefficients, whereas [MT] treats cases arising from C' metric tensors. This extends earlier work of [Ve] and others on the flat Laplacian on Lipschitz domains in Euclidean space. Prerequisites for this work include an acquaintance with basic results on pseu- dodifferential operators and some methods from harmonic analysis, including the Littlewood-Paley theory. Sufficient material on these prerequisites could be ob- tained from either [T2] or Chapters 7 and 13 of [T51. Indeed, this present work can be viewed as a companion to 1T2]. Michael Taylor
- 17. CHAPTER 1 Pseudodifferential Operators with Mildly Regular Symbols Introduction Studies of pseudodifferential operators whose symbols p(x, satisfy a Holder condition in x have been found to be very useful in PDE. A number of their prop- erties and applications are investigated in [Bon], [Bour], [KN], [Meyl], [T2], and other places. There has also been an interest in symbols whose i-dependence is described by some other modulus of continuity; studies of this are made in [CM], [Ma2], and [Ma3], for example. Here we study related problems. We concentrate on measures of regularity just a bit better than mere continuity, with a secondary interest in measures of regularity either barely better or barely worse than Lipschitz. We measure such regularity in several different ways. To be explicit, we consider four types of function spaces, associated with a modulus of continuity w. First, (0.1) Second, with A(j) = w(22), we say (0.2) E where is a Littlewood-Paley partition of unity and 14'j (D) the associated Fourier multiplier. Third, we say (0.3) u E IN' — <Cw(2 where 4,,(D), which is an approximate identity as j —. oo. Fourth, we say (0.4) e <C21w(23). When w(h) = h' (so A(j) = and r (0,1), then all the spaces (0.l)—(0.4) coincide, due to well known results of Bernstein, Jackson, and Zygmund. For more general r E R, (0.2) with A(j) = defines the Zygmund space C. For other interesting w, such as (0.5) w(h) = (for h E (0, 1/2]) with s > 0, or (0.6) w(h)
- 18. not all of these spaces coincide. In §1 we explore various properties of spaces of the form (0.1)—(0.4), including containment relations among them. In §2 we establish U-operator norm estimates for pseudodifferential operators (0.7) p(x,D)u= whose symbols p(x, 4) are of order 0 and satisfy various regularity conditions in x. We consider symbol classes of the form XSrO, where (0.8) p(x,4) e xsç"0 and X is some function space, such as etc. We take (4) — (1 + in all cases except when (4) occurs as an argument of the logarithm; then we take (4) = (4 + We also define in (2.52) symbol classes generalizing the Hörznander classes Sfl. We recall that (0.9) p(x,4) e sro We produce results in terms of and compare these with results of [CMI and [Ma3], given in terms of Neither set of results contains the other. We can slightly extend the scope of the latter class of estimates by recasting them in terms of In §3 we decompose a symbol, such as in (0.8), into a sum of two terms. One, p# (x, 4), has better smoothness properties, of use in further results on operator calculus. The remainder, p5(x, 4), has no better smoothness, but does have lower order (by a degree depending on the smoothness of p(x, 4)). Such symbol smooth- ings follow a number of earlier models, including [KN], [Bon], [Mey2], and [T2]. A particular case of fundamental importance gives rise to Bony's paraproduct. Constructions in §3 give rise to another class of symbol spaces, X defined in (3.18)—(3. 19). in §4 we analyze operators p(x, D) on variants of Sobolev spaces, of the form (0.10) = {f A(D)f E L"(lft't)}, where A(4) = (4)cL(t) is a symbol satisfying a certain "ellipticity" hypothesis. in §5 we treat operators p(x,D) on the spaces These are the spaces from the list (0. 1)—(0.4) that seem best suited for analysis involving pseudodiffer- ential operators. Largely for this reason, we have emphasized symbol spaces of the form c0')sr0 and over other symbol spaces. Indeed, the class of spaces C(A) is important even for analysis involving more usual sorts of pseudodifferential operators. We mention that, for slowly varying 0, (0.11) P e 0 S c 1 P: which is a consequence of Proposition 5.7. In §6 we study products of operators, particularly of the form p7 (x, D) where p7(x, 4) arises from p(x, 4) E XSr(k) via symbol smoothing. We apply this to commutator estimates in §7. in §8 we discuss results about operators whose symbols p(x, 4) have Sobolev space regularity in x.
- 19. I. In §9 we study operators arising from double symbols: (0.12) Au(x) = where a(x, y, has limited regularity in x and y. For example, parallel to (0.8), we say (0.13) a Banach space of functions on 1W' x 1W'. There are similar definitions of symbol classes X8r6, XSr(K)1 etc., for double symbols. While, as is well known, if E 8r6,o E [0,1), then A in (0.12) can be written in the form (0.7), with p(x, C such a reduction need not hold for symbol classes with limited regularity, so a separate study of operators of the form (0.12) is called for. One way operators of this sort arise is in the use of layer potential techniques for eliiptic PDE on domains whose boundaries have limited regularity (a little better than C'). We will explore this in §4 of Chapter III; this can be contrasted with the more difficult study of such PDE on Lipschitz domains, taken up in Chapter TV. In §10 we establish a commutator estimate of [CRW]. The approach we take uses paraproducts, and follows a proof given in [AT]. This commutator estimate plays an important role in div-curl lemmas, a topic that will be discussed in Chapter III. It also is a key tool in the analysis of operators with vmo coefficients, the subject of §11 of this chapter! In § 12 we investigate operators on the Besov spaces (1W'), defined by (0.14) 1 E B,,(1W') <cc. with as in (0.2). In particular, the space (1W') is of interest because it is contained in C (1W') but contains all spaces C(A)(R72) for which Tn §13 study a rather general class of operators with coefficients in a function algebra. One primary application of these results will be to cases involving Besov spaces that are algebras. In §14 we establish estimates on Pu, when P is a pseudodifferential operator and u is barely more regular than continuous, of a sort pioneered in [BKM]. We also produce improvements, involving estimates in and mention how such results relate to constructions in Chapter 5 of [T2]. In §15 we present some generalizations of an interesting inequality that arises in the study of CR manifolds in [Tu]. We interpret the estimates as commutator estimates. Tn §16 we look at some variants of Morrey spaces. We extend Morrey's imbed- ding theorem to a class of Morrey-type spaces. We discuss advantages of expressing these results in terms of spaces rather than the spaces more commonly used for such imbedding theorems. We also study the action of various pseudodif- ferential operators on these Morrey-type spaces. 1. Spaces of continuous functions Suppose w is a modulus of continuity, i.e., a continuous, increasing function on [0,00) such that w(0) = 0, and having the property that w does not vary too
- 20. rapidly, i.e., that w(2h) Cw(h), V h (0, 1J. We denote by CW the space of bounded, continuous functions on satisfying (1.1) u(x + y) - u(x)J 5; Cw(PyI). Next, if A(j) is a positive decreasing sequence, we want to define To do this, we use a Littlewood-Paley partition of unity. Pick C°° (RTh), so that Wo(E) = 1 for CI 5; 1, 0 for CI 2. Set (1.2) = = Wk(C) — for k 1. Also set øo(C) = 4'o(C), so (1.3) Now we say u E C(A) if and only if (1.4) 5; CA(j). We also make the standing assumption that is slowly varying, i.e., A(j) CA(j+1). PROPOSITION 1.1. If w is a modulus of continuity, then (1.5) A(j) = cw c PROOF. Assume u C Cw. Then, for 1, since = 0, = [u(x - y)- u(x)] dy 5; since is concentrated on yj $ (A more elaborate version of this argu- ment is given in (1.42)—(1.43) below.) Towards a converse, we have the following. PROpOSITION 1.2. If AU) is a positive decreasing sequence such that E Mi) < then (1.7) c(A) c cc, where (1.8) cr(h) = inf A(t) = 11(t) = >IMi). j'çe
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- 22. found. Practical instruction for their employment, as suggested by the writer’s own experience, have been added.” “Although there is necessarily a good deal of the descriptive catalogue in a work of this kind, yet this one is so well put together, its brief descriptions are so clear, and above all the endless varieties of tools enumerated are brought to one’s notice in so logical an order, their classification is so essentially scientific, that it may be regarded as in a sense a finished monograph of one phase of evolution.” + + + Ath. 1905, 2: 280. Ag. 26. 420w. * “The author has written a clear and comprehensive description of various groups of tools.” + Engin. N. 54:529. N. 16, ‘05. 120w. Hornung, Ernest William. Stingaree. †$1.50. Scribner. Stingaree, a one-time London clubman, now a robber in Australia, “sticks up” (Australian for hold up) mail coaches and banks in a manner both theatrical and gentlemanly. On one occasion he operates among a company of amateurs, forcing them to give a concert, and makes use of the occasion to introduce a girl with a beautiful voice to a prominent composer. He is afterward released from jail just in time to don evening clothes and hear this girl as a prima donna. “Of no importance from the literary standpoint, the present volume yet contains ten very readable and ingeniously worked out stories.” + — Ath. 1905, 1: 716. Je. 10, 180w.
- 23. “The stories are all fluent, ingenious, and diverting, and will be found readable enough.” + Critic. 47: 285. S. ‘05. 90w. “Series of ingenious tales.” + N. Y. Times. 10: 292. My. 6, ‘05. 250w. “On the whole, his adventures being as hazardous and exciting as those of his predecessor he should be equally well beloved.” + N. Y. Times. 10: 390. Je. 17, ‘05. 190w. “The tales ... are dashing, daring, entertaining, and show considerable inventiveness without disclosing any special literary power.” + Outlook. 80: 143. My. 13, ‘05. 160w. + Pub. Opin. 38: 869. Je. 3, ‘05. 210w. “He is a real creation.” + Reader. 6: 593. O. 05. 220w. + — R. of Rs. 31: 761. Je. ‘05. 90w. “Mr. Hornung who has much aptitude for sensational fiction has exhibited little ingenuity or originality in these tales.” + — Sat. R. 100: 26. Jl. 1, ‘05. 140w. * Hornung, Ernest William. Thief in the night: further adventures of A. J. Raffles, cricketer and cracksman. †$1.50. Scribner.
- 24. The third series of the adventures of Raffles goes back to the earliest days of the cracksman and Bunny, his foil. One of the nine tales portrays the disloyalty of the thief in losing for Bunny his sweetheart, another, and quite the most ingenious of the group, is that of a little “job” at Lord Thornaby’s town house where Raffles diverted from himself the suspicions of the “Criminologists’ club.” All thru Raffles is still the same terrible expert burglar. * + Acad. 68: 1177. N. 11, ‘05. 310w. * “Unfortunately the reader’s taste has been whetted for better things, and he looks in vain for the quick turns and the conquering of difficult situations of the earlier yarns.” + — Critic. 47: 578. D. ‘05. 70w. * “The newer stories, while they seem somehow to lack the snap and go of the earlier ones, are nevertheless not very different in quality, and if you are not tired of the old Raffles they may be trusted to furnish entertainment for an idle hour.” + N. Y. Times. 10: 727. O. 28, ‘05. 160w. * “Those unacquainted with the cracksman will find admirably written stories retailing the exploits of a gentleman burglar of the most marvelous skill and finesse, and an unusually winning personality.” + Outlook. 81: 530. O. 28, ‘05. 70w. * “His mind works with all its old rapidity and originality, but he is less convincing and beguiling.” + — Outlook. 81: 712. N. 25, ‘05. 100w. * “It is not so mischievous as its predecessors, because it is not nearly so well done.”
- 25. + — Spec. 95: 871. N. 25, ‘05. 90w. Horsley, Walter C., tr. See La Colonie, Jean Martin de. Hort, Fenton John Anthony. Village sermons. * $1.75. Macmillan. Dr. Hort, a noted scholar and Christian gentleman, writes with simplicity for the country folk with whom he had to deal as the parson of a Hertfordshire village. The sermons “are generally founded on some incident of the day’s service, some sentences in a psalm, or more often some petition in a collect.” (Lond. Times.) * + + Lond. Times. 4: 287. S. 8, ‘05. 470w. * + — Outlook. 81: 283. S. 30, ‘05. 70w. * “We must confess that the sermons strike us as being highly conscientious but a trifle dull. Yet here and there, genius shows itself in the easy power of expressing a great deal in a few words.” + — Sat. R. 100: 190. Ag. 5, ‘05. 140w. Horton, George. Monk’s treasure. $1.50. Bobbs. Ta Castra, an island of the Cyclades, in the Ægean sea, is the scene of a series of adventures in which a young American, buying up Greek argols for his uncle’s firm, and his interpreter, a sturdy Scotchman, figure conspicuously. The American
- 26. straightway becomes involved in breaking up an alliance between a beautiful Greek bond-girl and her belligerent betrothed, Spiro. Thru treasure, hidden in a monastery, he proves the girl to be a duchess, and outwitting the monks and Spiro alike, escapes with Polyxene and her bags of gold. “Those who love a story for the story’s sake will be sure to enjoy Mr. Horton’s latest romance.” Amy C. Rich. + + Arena. 33: 565. My. ‘05. 120w. “The recovery of the wealth against the cunning machinations of the monks supplies a number of exciting and tragic events to sustain interest in a story which otherwise is rather lightly worked out.” + — N. Y. Times. 10: 293. My. 6, ‘05. 280w. “Crude romance.” — Outlook. 79: 760. Mr. 25, ‘05. 80w. “A good story.” + R. of Rs. 31: 762. Je. ‘05. 70w. Hosking, Arthur Nicholas, comp. and ed. Artist’s year book. $3. Art league pub. assn., Chicago. A handy reference book wherein may be found interesting data pertaining to artists, and their studio, home and summer addresses for 1905-1906. Recognized merit has been made the standard of selection for this list. * Hough, Emerson. Heart’s Desire. †$1.50. Macmillan.
- 27. Heart’s Desire is a little settlement hidden away in a corner of the West “where men have gone to live at peace—without law and without women.” “The inhabitants dozed in the sunshine, smoked, drank, gambled a little, toiled fitfully, fought occasionally, and dreamed a good deal. Then the railroad came and the dreams were gone. Along with the railroad came Constance and the old vexations that troubled Eden and have troubled every assemblage of men ever since.” (Pub. Opin.) It is a picture of rough Western life with clever character delineation. * “A singularly pleasing story of the west o’ dreams.” + N. Y. Times. 10: 824. D. 2, ‘05. 150w. * “A more vivacious tale of far western life one does not often get.” + + Outlook. 81: 576. N. 4, ‘05. 100w. * “In vigor and spontaneousness it seems to us Mr. Hough’s best work in fiction.” + + Outlook. 81: 710. N. 25, ‘05. 120w. * “It is idyllic, impossible, and extremely entertaining.” + — Pub. Opin. 39: 699. N. 25, ‘05. 220w. Houston, Edwin James. Electricity in every-day life. 3v. $4.50. Collier. “These volumes aim to give to the general reader a comprehensive knowledge of the history of electricity, the principles and laws that govern its action, and its practical applications in every-day life.” (Outlook.) There are eight
- 28. hundred illustrations which present electricity as applied to modern industry and as used in laboratories, and in the home. “Without trace of romance and yet in an eminently attractive style, the author has made comparatively clear the vagaries of electricity.” + + Critic. 46: 384. Ap. ‘05. 90w. “The style is clear and pleasant. Abstruse technicalities are carefully avoided, and no part of the book will be difficult of comprehension for the average well-informed man who has made no specialty of electrical subjects.” + + N. Y. Times. 10: 276. Ap. 29, ‘05. 330w. Outlook. 79: 502. F. 25, ‘05. 30w. “He succeeds well in popularizing technical subjects. The present work is voluminous, but never wearisome.” + + R. of Rs. 31: 251. F. ‘05. 90w. Howard, George Elliott. History of matrimonial institutions chiefly in England and the United States. *$10. Univ. of Chicago press. “In the three volume work ... Prof. George E. Howard deals chiefly with the matrimonial institutions of the English race, prefacing his treatment of the subject with an analysis of the literature and the theories of primitive matrimonial institutions. Professor Howard’s treatise covers practically every phase of the subject that calls for treatment, and gives elaborate biographical data relating, not only to the institution of
- 29. marriage itself, but to almost every conceivable phase of the sex problem that has been treated in our literature.”—R. of Rs. “To students of sociology this work is one of importance.” Simeon E. Baldwin. + + Am. Hist. R. 10: 607. Ap. ‘05. 1070w. (Abstract of book). Reviewed by E. T. B. Atlan. 95: 137. Ja. ‘05. 650w. “Professor Howard’s volumes are admirable studies and a much needed supplement to the famous works of Starcke and Westermarck.” + + Ind. 58: 784. Ap. 6, ‘05. 720w. * “A scholarly and profound inquiry.” + + Ind. 59: 1158. N. 16, ‘05. 40w. “For even the general public Professor Howard’s volumes cannot fail to be both interesting and instructive, for they deal attractively with the most human of all institutions, and contain a mass of facts nowhere else obtainable.” + + + Nation. 80: 55. Ja. 19, ‘05. 2270w. R. of Rs. 30: 756. D. ‘05. 120w. “An immense bibliographical index at the end of the third volume completes the usefulness of the work as a book of reference, and it is as a book of reference that it will be chiefly used and valued.”
- 30. + + + Spec. 94: 617. Ap. 29, ‘05. 830w. Howard, John R., comp. See One hundred best American poems. Howe, Frederick Clemson. City: the hope * of democracy. **$1.50. Scribner. “A novel interpretation of municipal affairs.... Mr. Howe ascribes most of the ills to which the American city is heir to economic and industrial, rather than to political or ethical causes.... Mr. Howe’s remedy for the present evil conditions consists in offering opportunity to labor, in taxing monopoly, and in the abolition of privilege.”—R. of Rs. * “Mr. Howe’s main arguments in favor of municipal ownership are strong, and much of his abstract reasoning in favor of the single tax is well put, although less convincing to most people; but the author is too sweeping in his advocacy of the adoption of these measures and in his claims for resulting benefits.” + — Engin. N. 54: 648. D. 14, ‘05. 610w. * “Dr. Howe, in a spirited and striking description of the American city, interprets its myriad phases from the economic standpoint.” + Ind. 59: 1158. N. 16, ‘05. 20w. * “Is a good deal of a theorist, but, happily also, he is very much of a student. Mr. Howe’s book, we fear, will not advance that result as much as it might have done had it been more soberly written.” E. C.
- 31. + — N. Y. Times. 10: 773. N. 18, ‘05. 970w. * “His book is a frank discussion of municipal problems as they are actually encountered in the more typical of our American cities. The prevailing note is one of optimism.” + R. of Rs. 32: 637. N. ‘05. 230w. Howe, Maude. See Elliott, Mrs. Maude Howe. Howells, William Dean. London films. * **$2.25. Harper. The volume is made up of Mr. Howells’ characteristic talks about London weather, London streets, London noises, churches, parks, buses, slums, children, and bobbies—often with humorous comparison with the corresponding phenomena in New York. He tells, too, about society out of doors in Rotten Row and Piccadilly. Some of the matter has already appeared in some of the magazines. The book is provided with sixteen full-page illustrations and is bound to match the author’s “Literary friends and acquaintances.” * “These films do not amount to so comprehensive or extensive a survey as Emerson achieved. But they are very fascinating, and are written with the clarity and richness of style which constitute Mr. Howells one of our foremost writers of English to-day.” + + Ath. 1905, 2: 717. N. 25. 1740w. * “Its ‘films’ are far more interesting and significant than some that Mr. Howells has shown; they are indeed in his happiest analytic vein.”
- 32. + + Dial. 39: 381. D. 1, ‘05. 270w. * “The book is in no whit inferior to those masterly studies in Italian life.” + Ind. 59: 1227. N. 23, ‘05. 880w. * “A series of delicate and charming impressions of London in many of its aspects, social, civic, and meteorological.” + + Nation. 81: 490. D. 14, ‘05. 820w. * “Easily takes its place among the few most noteworthy books of the season.” + + N. Y. Times. 10: 712. O. 21, ‘05. 130w. * “He is still master of the gentle irony, the subtle, mischievous suggestion, the humorous backward glance, that have fascinated his readers for years.” + Outlook. 81: 682. N. 18, ‘05. 270w. Howells, William Dean. Miss Bellard’s inspiration. † $1.50. Harper. “It was nothing short of inspiration which made Miss Lillias Bellard decide to visit her aunt and uncle, the Crombies, in order to consider quietly the question of marrying a certain eager young Englishman. Mr. and Mrs. Crombie had ... taken a cottage in the New Hampshire hills. Miss Bellard’s intention was to watch the domestic conditions of the Crombie household before rushing recklessly into matrimony. But coincident with her visit came that of the Mevisons, a couple trembling upon the verge of separation. Thus Miss Bellard was
- 33. treated to a variety of domestic relations which produced varying effects upon her.”—N. Y. Times. “Charming and idyllic comedy which at once tickles and instructs. Mr. Howells has written no more delightful story for years.” + + Ath. 1905, 2: 41. Jl. 8, 310w. * “The book is undeniably a delicate and diverting piece of satire and full of those illuminating sidelights upon human foibles and frailties that make Mr. Howells inimitable.” Frederic Taber Cooper. + Bookm. 21: 610. Ag. ‘05. 1500w. * “The charm of Mr. Howells’s style is the only inducement offered the ‘gentle reader’ in this book.” Charlotte Harwood. + — Critic. 47: 452. N. ‘05. 240w. “Has a charm altogether out of proportion to its pretensions.” Wm. M. Payne. + + Dial. 39: 115. S. 1, ‘05. 280w. * Ind. 59: 1152. N. 16, ‘05. 60w. “The whole thing is dainty and amusing, and the irony so suavely expended that some readers may fail to detect it, and hence be a little puzzled as to the degree of the author’s facetiousness.” + + + Lit. D. 31: 187. Ag. 5, ‘05. 790w.
- 34. “It is as if Mr. Howells’s vision were being contracted instead of enlarged as the years go on. He stops short now at the surface; and delicately and gracefully as he plays about on it, we regret his arrested development.” + — Lond. Times. 4: 209. Je. 30, ‘05. 520w. “Is a light comedy with enough social satire to remind us that Mr. Howells is not just fooling for our summer holiday.” + + Nation. 81: 101. Ag. 3, ‘05. 320w. “Mr. Howells has not lost any of his cunning in portraying the delightfully illogical phases of the feminine mental processes. Altogether it is a decidedly entertaining book.” + + N. Y. Times. 10: 389. Je. 17, ‘05. 160w. “Has all the lightness, the charming comedy touch, of his earlier work, and yet is not lacking in serious purpose. The studies of temperament are both skillful and convincing. It is quite certain that Mr. Howells has written nothing in a happier style; the vein of humor which runs through the book is as fresh as in his earlier work, and parallel with it runs a vein of quiet, kindly irony equally effective.” + + + Outlook. 80: 643. Jl. 8, ‘05. 190w. “Beyond a doubt the story is amusing, but to Mr. Howells’ real devotees it must be rather hard sledding.” + — Pub. Opin. 39: 283. Ag. 26, ‘05. 650w.
- 35. * “Though but a slight love tale, embodies a maturity of conception, a surety of view, a subtle phraseology, an exquisite use of irony, and, withal, a sedate, appeasing dignity.” + R. of Rs. 32: 757. D. ‘05. 70w. “The book is mainly a study—and a very clever and shrewd study—of one type of American girl. But all the subordinate characters are carefully drawn.” + + — Spec. 95: 124. Jl. 22, ‘05. 830w. Howells, William Dean. Son of Royal Langbrith. $2. Harper. The story is the tragic one of the weakness of a good mother who lacks the courage to tell her son of the iniquities of his dead father. He grows up in the belief that his father is a noble and heroic character, and when the truth is revealed to him, through the courtship of his mother by the country doctor, he suffers greatly in the loss of his ideal. An opium eater and his loyal daughter enter into the story. The setting is a small New England manufacturing town. “Is in many respects the best bit of work Mr. Howells has done of late years. One is inclined to read it slowly, lingering in enjoyment of the charming style, and appreciating to the full the perfect picture of New England life in the minute details that Mr. Howells so loves to dwell on. It is a pity, however, that in his love of realistic detail, Mr. Howells should be led into writing passages which, to say the least, mar the artistic effect of his work. He has set such a dainty dish before us that we cannot bear even one drop of grease to spoil the taste.” C. Harwood. + + Critic. 46: 184. F. ‘05. 560w.
- 36. — “The one objection which the average reader has been known to make against the work of Mr. William Dean Howells,— namely, that that distinguished novelist is too fond of the insignificant,—cannot be brought against ‘The son of Royal Langbrith.’ That the working out of this theme is masterly it is superfluous to add.” + + R. of Rs. 31: 116. Ja. ‘05. 80w. “What lends peculiar charm to Mr. Howells’s best work is the fact that it could only have been written by an American. It is in the delicacy and tact with which it is hand sovereign merit of the story resides.” + + Spec. 94: 22. Ja. 7, ‘05. 960w. Hubbard, Arthur John, and Hubbard, George. Neolithic dew-ponds and cattle-ways. *$1.25. Longmans. “The author endeavors to solve the question of the water- supply of the Neolithic dwellers in hill-encampments on the downs in the south of England. There were apparently no wells, and they had to depend on the ‘unfed’ artificial dew- pond.... Closely connected with the dew-ponds are the cattle- ways down which primitive man drove his herds from the entrenched settlement to water.... There are numerous and very clear photographs.”—Nation. “Altogether the book is one to be read with interest and profit by everyone at all interested in the evidences relating to our ancestors of the stone age.” + + Ann. Am. Acad. 25: 589. My. ‘05. 220w.
- 37. “Contains much suggestive and interesting matter, and is very good reading, but not wholly convincing.” + — Ath. 1905, 2: 151. Jl. 29, 1080w. “The whole study is well worth reading even by those who have no immediate interest in antiquarian topography.” + + Nation. 80: 360. My. 4, ‘05. 830w. “The construction of dew-ponds by the early inhabitants of Britain has often been glibly asserted, but few, if any, have furnished such clear and circumstantial evidence as the authors of this short treatise.” + + + Nature. 71: 611 Ap. 27, ‘05. 610w. Hubbard, Gardiner Greene. Collection of engravings. See United States, Library of Congress. Hubbard, Sarah A., comp. See Catch words of cheer. Huckel, Oliver., tr. Lohengrin, **75c. Crowell. A companion volume to Mr. Huckel’s “Parsifal” which appeared in similar form two years ago. “It is a version for the general reader. It is not a libretto for the music. It gives a cumulative impression, the composite effect of words, scenery, action, and it is hoped, the spirit of the musical interpretation ... the spirit of the original text in a free version rather than in a strictly literal one.”
- 38. * “The poem is preceded by an admirable introductory chapter relating to the work, the whole forming a little volume which will be highly prized by lovers of this noble music-drama.” + Arena. 34: 557. N. ‘05. 140w. “It gives the reader a much better impression of the drama than the ordinary literally translated libretto can furnish.” + Outlook. 81: 629. N. 11, ‘05. 40w. * “The verse is smooth and dignified.” + R of Rs. 32: 751. D. ‘05. 30w. Huckel, Oliver. Melody of God’s love; a new unfolding of the twenty-third psalm, *75c. Crowell. An interpretation of the twenty-third psalm which divides it into three melodies: In green pastures, a song of the sweet and pleasant experiences of life; Through the valley of the shadow, a song of the harder and deeper and more sorrowful experiences of life; and, In the house of the Lord forever, a song of the exultant and triumphant and heavenly experiences of life here and hereafter. “A series of meditative essays in poetic vein, but without great distinction of style.” + — Outlook. 81: 629. N. 11, ‘05. 20w. Huffcut, Ernest Wilson. Elements of business law; with illustrative examples and problems. *$1. Ginn. This volume is intended as a text-book for students in commercial courses in high schools and colleges and it sets forth the fundamental principles of business law, giving simple
- 39. concrete examples which show them in their actual application to business transactions. Problems taken from decided cases are given at the end of each chapter. The book is based upon the common law and a glossary of legal terms is provided.
- 40. Hughes, Hugh Price. Life of Hugh Price Hughes, by his daughter. 3d ed. *$3. Armstrong. “Mr. Price Hughes broke in early life with the traditional conservatism of the Methodist body, and allied himself with the Liberation society.... The greater part of the volume is taken up with the spiritual activities with which” he “occupied his strenuous life. These were very various in kind. Not the least interesting among them is the part which he took in the reunion conferences at Grindelwald.”—Spec. “She tries to set down all her father ever did or said, with little order of time and not too much of logic; yet large abstractions obscure practical details.” — + Nation. 80: 354. My. 4, ‘05. 630w. N. Y. Times. 10: 196. Ap. 1, ‘05. 480w. “This story of his life will be read in all branches of the Church. It deserves to be. It needs to be.” + + + Outlook. 79: 143. Ja. 14, ‘05. 270w. “We must frankly say that there is a certain magniloquence of diction and general exuberance about Miss Hughes’s description of her father’s life and work which we could wish away; but these do not hinder us from recognizing a really striking personality. There are, indeed, more serious faults in Miss Hughes’s book than those of diction and manner. It would not have cost much trouble to ascertain the facts.” + — Spec. 94: 181. F. 4, ‘05. 520w.
- 41. * Hughes, Rupert. Zal: an international romance. † $1.50. Century. The tale of a young Polish pianist’s battle for recognition in New York. There is the artist and dreamer’s “deathless enthusiasm” which dominates Ladislav Moniusko and Rose Hargrave, a wealthy New York girl, whose father had set her apart for an English duke. * “The book is of value, not only because of its musical quality, but because it enlarges information and intensifies sympathy for what may truly be called the land of genius.” + N. Y. Times. 10: 856. D. 2, ‘05. 630w. * “The contrast between the Polish and American natures is excellently indicated.” + Outlook. 81: 838. D. 2, ‘05. 70w. Hugo, Victor. Notre Dame de Paris. $1.25. Crowell. A volume in the “Thin paper classics” series, translated from the French by Isabel F. Hapgood. Hugo, Victor. Toilers of the sea. $1.25. Crowell. A translation from the French by Isabel F. Hapgood, uniform with the other attractive volumes of the “Thin paper classics” series. Hulbert, Archer Butler, and others. Future of road making in America. (Historic highways in America.) *$2.50. Clark, A. H. Volume XV closes the series of monographs on the history of America as portrayed in the evolution of its highways of war, commerce, and social expansion, in the “Historic highways of America” series. Besides the first essay, which gives the title to
- 42. this volume, Mr. Hulbert’s symposium includes “Government co-operation in object-lesson road work,” by Martin Dodge; Maurice O. Eldridge’s “Good roads for farmers,” Prof. Logan Waller Page’s “The selection of materials for macadam roads,” and E. G. Harrison’s “Stone roads in New Jersey.” There will be a final volume devoted to an index. + — Am. Hist. R. 10: 928. Jl. ‘05. 110w. “The later volumes of the series present both the merits and defects of the earlier ones. They are entertaining and often suggestive, but always incomplete. The material is ill arranged, and a surprising amount of it is reprinted from other books.” + — Dial. 38: 322. My. 1, ‘05. 290w. (Review of v. 15 and 16.) + + Engin. N. 53: 183. F. 16, ‘05. 300w. N. Y. Times. 10: 132. Mr. 4, ‘05. 390w. “A model of what an index should be.” + + + N. Y. Times. 10: 514. Ag. 5, ‘05. 140w. (Review of v. 16.) + + Outlook. 79: 449. F. 18, ‘05. 120w. “[The index] is model work of its kind.” + + Pub. Opin. 38: 838. My. 27, ‘05. 120w. (Review of v. 15 and 16.)
- 43. Hume, Fergus. Mandarin’s fan. †$1.25. Dillingham. A jade fan causes a deal of trouble in Mr. Hume’s new story. It is at the bottom of a plot which involves the good name of a Chinese official, the fortunes of a worthy young Englishman, the satisfaction of a Chinese god, Kwang-Ho, and the happiness of a young English girl. Never has the author presented so motley an array of men and women from which to select the real criminal. “His ‘heathen’ are of the conventional and traditional sort, but the dialogue is spicy, the plot intricate, and the personages are set in lively contrast to each other.” + — Critic. 47: 189. Ag. ‘05. 70w. “Clever as the plot is, there are several woefully weak links, though a rapid reader is pretty sure to overlook these while engrossed in the really thrilling story. Contrary to his usual custom, Fergus Hume has given us better character drawing than plot in this tale of a fan.” + — N. Y. Times. 10: 262. Ap. 22, ‘05. 230w. “The author appears frequently quite as much in the dark in trying to fix on one of his puppets the crime involved as is the patient and mystified reader. There is here no real flesh and blood.” — Outlook. 79: 706. Mr. 18, ‘05. 70w. Hume, Fergus W. Secret passage. †$1.25. Dillingham. The secret passage contains all the strange things which are the natural accompaniments of secret passages. An eccentric old lady is found stabbed to death in her room and there is no clue to the murderer. A clever young detective takes up the
- 44. case and a number of people become involved in it; several love stories past and present serve to make matters more complicated, and in the end it is discovered that the murdered old lady was really somebody else in disguise, and that the only person not suspected of the crime is the guilty one. “Another of his hide-and-seek, jack-o’-lantern murder mysteries.” + N. Y. Times. 10: 589. S. 9, ‘05. 230w. Hume, Martin Andrew S. Spanish influence of English literature. $2. Lippincott. Ten public lectures re-written fill this portly volume. Its aim is “to provide for English readers a comparative study of Spanish literature in special relation to its points of contact with the literature of our own country.” “Major Hume does not succeed in persuading us that he has attained to any clear conception of what is meant by literary influence.” — Acad. 68: 99. F. 4, ‘05. 840w. “But these faults of arrangement, selection, and taste are minor defects in comparison with the want of knowledge and the inaccuracy which the book shows. Instances of reckless assertion are numerous in every chapter.” — — — Ath. 1905, 1: 365. Mr. 25, ‘05. 1200w. “The book, moreover, seems to be pervaded by an exaggerated sense of the importance of its thesis.” — Dial. 39: 93. Ag. 16, ‘05. 240w.
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