© 2016 Gurbuz, published by De Gruyter Open.
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Open Math. 2016; 14: 300–323
Open Mathematics Open Access
Research Article
Ferit Gurbuz*
Parabolic sublinear operators
with rough kernel generated by parabolic
Calderón-Zygmund operators and parabolic
local Campanato space estimates for their
commutators on the parabolic generalized
local Morrey spaces
DOI 10.1515/math-2016-0028
Received February 2, 2016; accepted April 25, 2016.
Abstract: In this paper, the author introduces parabolic generalized local Morrey spaces and gets the boundedness of
a large class of parabolic rough operators on them. The author also establishes the parabolic local Campanato space
estimates for their commutators on parabolic generalized local Morrey spaces. As its special cases, the corresponding
results of parabolic sublinear operators with rough kernel and their commutators can be deduced, respectively. At
last, parabolic Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an
example.
Keywords: Parabolic singular integral operator, Parabolic sublinear operator, Parabolic maximal operator, Rough
kernel, Parabolic generalized local Morrey space, Parabolic local Campanato spaces, Commutator
MSC: 42B20, 42B25, 42B35
1 Introduction
Let R
n
be the ndimensional Euclidean space of points x D .x
1
; :::; x
n
/ with norm jxjD
n
P
i D1
x
2
i
! 1
2
. Let
B D B.x
0
;r
B
/ denote the ball with the center x
0
and radius r
B
. For a given measurable set E, we also denote
the Lebesgue measure of E by jEj. For any given R
n
and 0<p< 1, denote by L
p
./ the spaces of all
functions f satisfying
kf k
L
p
./
D
0
@
Z
jf .x/j
p
dx
1
A
1
p
< 1:
Let S
n1
Dfx 2 R
n
WjxjD 1g denote the unit sphere on R
n
.n 2/ equipped with the normalized Lebesgue
measure d .x
0
/, where x
0
denotes the unit vector in the direction of x.
*Corresponding Author: Ferit Gurbuz: Ankara University, Faculty of Science, Department of Mathematics, Tando˘ gan 06100,
Ankara, Turkey, E-mail: feritgurbuz84@hotmail.com
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