Georgian Math. J. 20 (2013), 805 – 816
DOI 10.1515 / gmj-2013-0037 © de Gruyter 2013
About integral operators of fractional type
on variable L
p
spaces
Pablo Rocha and Marta Urciuolo
Abstract. If A
1
;:::;A
m
are orthogonal matrices, in this paper we obtain strong and weak
estimates for certain integral operators with kernels of the form k
1
.x A
1
y/ k
m
.x
A
m
y/, in the context of Lebesgue spaces with variable exponents.
Keywords. Variable exponents, fractional operators.
2010 Mathematics Subject Classification. 42B25, 42B35.
1 Introduction
Given a measurable set R
n
, and a measurable function p./ W ! Œ1; 1/,
let L
p./
./ denote the Banach space of measurable functions f on such that
for some >0,
Z
jf.x/j
p.x/
dx < 1;
with the norm
kf k
p./
D inf
°
>0 W
Z
jf.x/j
p.x/
dx 1
±
:
These spaces are known as variable exponent spaces and are a generalization of the
classical Lebesgue spaces L
p
./. In the last years many authors have extended
the machinery of classical harmonic analysis to these spaces. The first step was
to determine sufficient conditions on p./ for the boundedness on L
p./
of the
Hardy–Littlewood maximal operator
Mf .x/ D sup
B
1
jB j
Z
B\
jf.y/jdy;
where the supremum is taken over all balls B containing x. Let p
D ess inf p.x/
and p
C
D ess sup p.x/. In [4], Cruz-Uribe, Fiorenza and Neugebauer prove the
following result.
The authors are partially supported by SECYTUNC and Conicet.
Brought to you by | University of Calgary
Authenticated
Download Date | 5/26/15 3:08 PM