Hardy-Steklov operator on two exponent Lorentz spaces for non-decreasing functions
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The document presents theorems characterizing when the Hardy-Steklov operator is bounded from one two-exponent Lorentz space to another. Specifically, it provides conditions on weights v and w such that the operator is bounded from L(0,∞)qpv to L(0,∞)srw. It defines the Hardy-Steklov operator and two-exponent Lorentz spaces. It states two theorems that characterize the weights using inequalities involving the weights and derivatives of the functions defining the Hardy-Steklov operator. The theorems assume the functions satisfy certain conditions like being strictly increasing and having derivatives satisfying an inequality.
is finite.
In this paper, we characterize the weights v and w for which a constant 0>C exists such that
0,,, ffCTf qp
vLsr
wL
(2)
where T is the Hardy-Steklov operator defined as
.=))(( )(
)(
)(
txTf dtf
xb
xa (3)](https://image.slidesharecdn.com/l011118386-151123084219-lva1-app6892/85/Hardy-Steklov-operator-on-two-exponent-Lorentz-spaces-for-non-decreasing-functions-1-320.jpg)
]([
=
1/
0>
1/
/1
0
,
stt
sdttst
f
rv
f
t
s
rsv
f
s
sr
vL
(4)
Proof. Applying the change of variable )(= ty v
f to the R.H.S. of (1) and integrating by parts we
get the lemma.
Lemma 2. If f is nonnegative and non-decreasing, then
.)()()(=
1
0
, dxxvdttvxf
r
s
f r
s
x
ss
sr
vL
(5)
Proof. We obtain the above equality by evaluating the two iterated integrals of )()(
1
1
xvxh
r
s
st r
s
s
over the set }<0),(<<0;),{( xxfttx , so that we have
dxdtxvxh
r
s
st r
s
sxf
)()(
1
1)(
00
,)()(=
1
1
)(0
dtdxxvxh
r
s
st r
s
s
tx
(6)
where })(:{sup=)( txfxtx for a fixed t , and .)(=)( dttvxh
x
Integrating with respect to ‘t ’ first, the L.H.S. of (6) gives us the R.H.S. of (5). Further
rs
tx
r
s
r
s
tx
dssvtxhdxxvxh
r
s /
)(
1
)(
)(=))((=)()(
.)]([=}]>)(:{[= r
s
v
f
r
s
ttxfxv
Hence the lemma now follows in view of Lemma 1.
III. Main Results
Theorem 1. Let <,,,<0 srqp be such that .<<1 sq Let T be the Hardy-Steklov operator
given in (3) with functions a and b satisfying the conditions given thereat. Also, we assume that )(<)( xbxa
for ).(0,x Then the inequality
q
pqq
v
s
rss
w
x
dx
xxf
p
q
C
x
dx
xxTf
r
s
1/
/*
0
1/
/*
0
)]([)]([
(7)](https://image.slidesharecdn.com/l011118386-151123084219-lva1-app6892/85/Hardy-Steklov-operator-on-two-exponent-Lorentz-spaces-for-non-decreasing-functions-2-320.jpg)
![Hardy-Steklov operator on two exponent Lorentz spaces for non-decreasing functions
DOI: 10.9790/5728-11118386 www.iosrjournals.org 85 | Page
holds for all nonnegative non-decreasing functions f if and only if
s
r
s
y
x
t
tbxa
xt
dyywdzzw
r
s
1/
1
)(<)(
<<<0
)()(sup
.<)()(
1/1
1)(
)(
qq
p
q
y
tb
xa
dyyvdzzv (8)
Proof. Using differentiation under the integral sign, the condition )(<)( xbxa for )(0,x ensures
that Tf is nonnegative and non-decreasing. Consequently, by Lemma 2, the inequality (7) is equivalent to
q
q
ssxb
xa
dxxVxfCdxxWdttf
1/
0
1/
)(
)(0
)()()()(
(9)
where )()(=)(
1
xwdzzw
r
s
xW r
s
x
and ).()(=)(
1
xvdzzv
p
q
xV p
q
x
Thus it suffices to show that (9) holds if and only if (8) holds. The result now follows in view of
Theorem 3.11 [2].
Similarly, in view of Theorem 2.5 [1], by making simple calculations, we may obtain the following:
Theorem 2. Let <,,,<0 srqp be such that .<<,1<<0 qqs Let T be the Hardy-Steklov
operator given in (3) with functions a and b satisfying the conditions given thereat. Also, we assume that
)(<)( xbxa for ).(0,x Then the inequality (7) holds for all nonnegative non-decreasing functions f
if and only if
plpqpqplrsrst
tab
txxbta //////
)((10
][)]()([
<)()()(
1/
1
l
p
q
x
dttdxxvdyyv
p
q
and
plpqpqplrsrstba
t
xttbxa //////))((1
0
][)]()([ ,<)()()(
1/
1
l
p
q
x
dttdxxvdyyv
p
q
where ,
11
=
1
,
11
=
1
qslpqr
and is the normalizing function as defined in [3].
Remark. The condition )(<)( xbxa for )(0,x cannot be relaxed since otherwise the monotonicity of
Tf would be on stake. For example, consider the functions
20
,1)29(
10
20<10
,910
10<0,
10
=)(
xx
x
x
x
x
xa
and](https://image.slidesharecdn.com/l011118386-151123084219-lva1-app6892/85/Hardy-Steklov-operator-on-two-exponent-Lorentz-spaces-for-non-decreasing-functions-3-320.jpg)
![Hardy-Steklov operator on two exponent Lorentz spaces for non-decreasing functions
DOI: 10.9790/5728-11118386 www.iosrjournals.org 86 | Page
20.
,1)299(1010
20<10
,99
10
10<0,1010
=)(
x
x
xx
xx
xb
Note that a and b satisfy all the aforementioned conditions, except that, we have )(>)( xbxa for
20<10 x .
Acknowledgement
This work was supported in part by the Department of Science and Technology (DST), INDIA.
References
[1]. H.P.Heinig and G.J. Sinnamon, Mapping properties of integral averaging operators, Studia Math., 129(1998), 157-177.
[2]. A. Kufner and L.E. Persson, Weighted Inequalites of Hardy Type, World Scientific Publishing Co. Pte. Ltd, Singapore, 2003.
[3]. E. Sawyer, Weighted Lebesgue and Lorentz norm inequalites for the Hardy operator, Trans. of Amer. Math. Soc., Vol. 281(1984),
329-337 .](https://image.slidesharecdn.com/l011118386-151123084219-lva1-app6892/85/Hardy-Steklov-operator-on-two-exponent-Lorentz-spaces-for-non-decreasing-functions-4-320.jpg)
Hardy-Steklov operator on two exponent Lorentz spaces for non-decreasing functions
- 1. IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. 1 (Jan - Feb. 2015), PP 83-86 www.iosrjournals.org DOI: 10.9790/5728-11118386 www.iosrjournals.org 83 | Page Hardy-Steklov operator on two exponent Lorentz spaces for non-decreasing functions Arun Pal Singh Department of Mathematics, Dyal Singh College (University of Delhi) Lodhi Road, New Delhi - 110 003, INDIA Email: arunps12@yahoo.co.in Abstract: In this paper, we obtain the characterization on pair of weights v and w so that the Hardy-Steklov operator dttf xb xa )( )( )( is bounded from )(0,, qp vL to )(0,, sr wL for <,,,<0 srqp . 2010 AMS Mathematics Subject Classification: 26D10, 26D15. Keywords: Hardy-Steklov operator, Lorentz spaces, non-decreasing. I. Introduction By a weight function u defined on )(0, we mean a non-negative locally integrable measurable function. We take ))(),((0,00 dxxu MM to be the set of functions which are measurable, non-negative and finite a.e. on )(0, with respect to the measure dxxu )( . Then the distribution function u f of 0Mf is given by .0,)(=:)( }>)(:)(0,{ tdxxut txfx u f The non-increasing rearrangement * uf of f with respect to )(xdu is defined as 0.},)(:{inf=:)(* yyttyf u fu For qp <0,<<0 , the two exponent Lorentz spaces )(0,, qp vL consist of 0Mf for which =,)(*1/sup 0> <<0, 1/ )](*1/[ 0, =: qtvfpt t q q t dtqtvfpt p q qp vL f (1) is finite. In this paper, we characterize the weights v and w for which a constant 0>C exists such that 0,,, ffCTf qp vLsr wL (2) where T is the Hardy-Steklov operator defined as .=))(( )( )( )( txTf dtf xb xa (3)
- 2. Hardy-Steklov operator on two exponent Lorentz spaces for non-decreasing functions DOI: 10.9790/5728-11118386 www.iosrjournals.org 84 | Page The functions )(= xaa and )(= xbb in (3) are strictly increasing and differentiable on )(0, . Also, they satisfy .<<0for)(<)(and=)(=)(0;=(0)=(0) xxbxababa Clearly, 1 a and 1 b exist, and are strictly increasing and differentiable. The constant C attains different bounds for different appearances. II. Lemmas Lemma 1. We have .=,)]([sup <<0,)]([ = 1/ 0> 1/ /1 0 , stt sdttst f rv f t s rsv f s sr vL (4) Proof. Applying the change of variable )(= ty v f to the R.H.S. of (1) and integrating by parts we get the lemma. Lemma 2. If f is nonnegative and non-decreasing, then .)()()(= 1 0 , dxxvdttvxf r s f r s x ss sr vL (5) Proof. We obtain the above equality by evaluating the two iterated integrals of )()( 1 1 xvxh r s st r s s over the set }<0),(<<0;),{( xxfttx , so that we have dxdtxvxh r s st r s sxf )()( 1 1)( 00 ,)()(= 1 1 )(0 dtdxxvxh r s st r s s tx (6) where })(:{sup=)( txfxtx for a fixed t , and .)(=)( dttvxh x Integrating with respect to ‘t ’ first, the L.H.S. of (6) gives us the R.H.S. of (5). Further rs tx r s r s tx dssvtxhdxxvxh r s / )( 1 )( )(=))((=)()( .)]([=}]>)(:{[= r s v f r s ttxfxv Hence the lemma now follows in view of Lemma 1. III. Main Results Theorem 1. Let <,,,<0 srqp be such that .<<1 sq Let T be the Hardy-Steklov operator given in (3) with functions a and b satisfying the conditions given thereat. Also, we assume that )(<)( xbxa for ).(0,x Then the inequality q pqq v s rss w x dx xxf p q C x dx xxTf r s 1/ /* 0 1/ /* 0 )]([)]([ (7)
- 3. Hardy-Steklov operator on two exponent Lorentz spaces for non-decreasing functions DOI: 10.9790/5728-11118386 www.iosrjournals.org 85 | Page holds for all nonnegative non-decreasing functions f if and only if s r s y x t tbxa xt dyywdzzw r s 1/ 1 )(<)( <<<0 )()(sup .<)()( 1/1 1)( )( qq p q y tb xa dyyvdzzv (8) Proof. Using differentiation under the integral sign, the condition )(<)( xbxa for )(0,x ensures that Tf is nonnegative and non-decreasing. Consequently, by Lemma 2, the inequality (7) is equivalent to q q ssxb xa dxxVxfCdxxWdttf 1/ 0 1/ )( )(0 )()()()( (9) where )()(=)( 1 xwdzzw r s xW r s x and ).()(=)( 1 xvdzzv p q xV p q x Thus it suffices to show that (9) holds if and only if (8) holds. The result now follows in view of Theorem 3.11 [2]. Similarly, in view of Theorem 2.5 [1], by making simple calculations, we may obtain the following: Theorem 2. Let <,,,<0 srqp be such that .<<,1<<0 qqs Let T be the Hardy-Steklov operator given in (3) with functions a and b satisfying the conditions given thereat. Also, we assume that )(<)( xbxa for ).(0,x Then the inequality (7) holds for all nonnegative non-decreasing functions f if and only if plpqpqplrsrst tab txxbta ////// )((10 ][)]()([ <)()()( 1/ 1 l p q x dttdxxvdyyv p q and plpqpqplrsrstba t xttbxa //////))((1 0 ][)]()([ ,<)()()( 1/ 1 l p q x dttdxxvdyyv p q where , 11 = 1 , 11 = 1 qslpqr and is the normalizing function as defined in [3]. Remark. The condition )(<)( xbxa for )(0,x cannot be relaxed since otherwise the monotonicity of Tf would be on stake. For example, consider the functions 20 ,1)29( 10 20<10 ,910 10<0, 10 =)( xx x x x x xa and
- 4. Hardy-Steklov operator on two exponent Lorentz spaces for non-decreasing functions DOI: 10.9790/5728-11118386 www.iosrjournals.org 86 | Page 20. ,1)299(1010 20<10 ,99 10 10<0,1010 =)( x x xx xx xb Note that a and b satisfy all the aforementioned conditions, except that, we have )(>)( xbxa for 20<10 x . Acknowledgement This work was supported in part by the Department of Science and Technology (DST), INDIA. References [1]. H.P.Heinig and G.J. Sinnamon, Mapping properties of integral averaging operators, Studia Math., 129(1998), 157-177. [2]. A. Kufner and L.E. Persson, Weighted Inequalites of Hardy Type, World Scientific Publishing Co. Pte. Ltd, Singapore, 2003. [3]. E. Sawyer, Weighted Lebesgue and Lorentz norm inequalites for the Hardy operator, Trans. of Amer. Math. Soc., Vol. 281(1984), 329-337 .