Research Article
Riemann-Liouville and Higher Dimensional Hardy Operators
for NonNegative Decreasing Function in
(⋅)
Spaces
Muhammad Sarwar,
1
Ghulam Murtaza,
2
and Irshaad Ahmed
2
1
Department of Mathematics, University of Malakand, Chakdara, Lower Dir 18800, Pakistan
2
Department of Mathematics, GC University, Faisalabad, Faisalabad 38000, Pakistan
Correspondence should be addressed to Muhammad Sarwar; sarwarswati@gmail.com
Received 23 May 2014; Accepted 6 August 2014; Published 26 August 2014
Academic Editor: Norio Yoshida
Copyright © 2014 Muhammad Sarwar et al. his is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
One-weight inequalities with general weights for Riemann-Liouville transform and -dimensional fractional integral operator in
variable exponent Lebesgue spaces deined on R
are investigated. In particular, we derive necessary and suicient conditions
governing one-weight inequalities for these operators on the cone of nonnegative decreasing functions in
()
spaces.
1. Introduction
We derive necessary and suicient conditions governing the
one-weight inequality for the Riemann-Liouville operator
()=
1
∫
0
()
(−)
1−
0<<1 (1)
and -dimensional fractional integral operator
()=
1
||
∫
||<||
()
|−|
−
0<<, (2)
on the cone of nonnegative decreasing function in
()
spaces.
In the last two decades a considerable interest of
researchers was attracted to the investigation of the mapping
properties of integral operators in so-called Nakano spaces
(⋅)
(see, e.g., the monographs [1, 2] and references therein).
Mathematical problems related to these spaces arise in
applications to mechanics of the continuum medium. For
example, Ruˇ zicka [3] studied the problems in the so-called
rheological and electrorheological luids, which lead to spaces
with variable exponent.
Weighted estimates for the Hardy transform
(
1
)()=∫
0
(), >0, (3)
in
(⋅)
spaces were derived in the papers [4] for power-
type weights and in [5–9] for general weights. he Hardy
inequality for nonnegative decreasing functions was studied
in [10, 11]. Furthermore Hardy type inequality was studied
in [12, 13] by Rafeiro and Samko in Lebesgue spaces with
variable exponent.
Weighted problems for the Riemann-Liouville transform
in
()
spaces were explored in the papers [5, 14–16] (see also
the monograph [17]).
Historically, one and two weight Hardy inequalities on the
cone of nonnegative decreasing functions deined on R
+
in
the classical Lebesgue spaces were characterized by Arino and
Muckenhoupt [18] and Sawyer [19], respectively.
It should be emphasized that the operator
() is the
weighted truncated potential. he trace inequity for this
operator in the classical Lebesgue spaces was established by
Sawyer [20] (see also the monograph [21], Ch.6 for related
topics).
In general, the modular inequality
∫
1
0
∫
0
()
()
V ()≤∫
1
0
()
()
()
(∗)
for the Hardy operator is not valid (see [22], Corollary 2.3,
for details). Namely, the following fact holds: if there exists
a positive constant such that inequality (∗) is true for all
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2014, Article ID 621857, 5 pages
http://dx.doi.org/10.1155/2014/621857