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.

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 .