Boundedness of multilinear singular integrals on central Morrey spaces with variable exponents
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Boundedness of multilinear singular integrals on central Morrey spaces with variable exponents Hongbin WANG1,2 , Jingshi XU3 ,
Jian TAN4
1 School of Mathematical Sciences, Anhui University, Hefei 230601, China 2 School of Mathematics and Statistics, Shandong University of Technology, Zibo 255000, China 3 School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China 4 School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China
c Higher Education Press 2020
Abstract We prove the boundedness for a class of multi-sublinear singular integral operators on the product of central Morrey spaces with variable exponents. Based on this result, we obtain the boundedness for the multilinear singular integral operators and two kinds of multilinear singular integral commutators on the above spaces. Keywords Multilinear singular integral, variable exponent, central Morrey space MSC2020 42B20, 42B35, 46E30 1
Introduction
Function spaces with variable exponents are being concerned with strong interest not only in harmonic analysis but also in applied mathematics. In the past several decades, the theory of function spaces with variable exponents has made great progress since some elementary properties were given by Kov´aˇcik and R´akosn´ık [18]. Lebesgue and Sobolev spaces with integrability exponent have been widely studied, see [4,7] and references therein. In [5,6,10] and [30– 34], the authors proved the boundedness of some integral operators on variable exponent function spaces, respectively. Many applications of these spaces were given, for example, in the modeling of electrorheological fluids [23], in the study of image processing [2], and in differential equations with nonstandard growth Received June 23, 2020; accepted September 24, 2020 Corresponding author: Hongbin WANG, E-mail: [email protected]
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Hongbin WANG et al.
[14]. First, we recall some notations and basic definitions on variable exponent Lebesgue spaces. Let E be a measurable set in Rn with |E| > 0. Given a measurable function p(·) : E → [1, ∞), p0 (·) is the conjugate exponent defined by p(·) p0 (·) = . p(·) − 1 The set P(E) consists of all p(·) : E → [1, ∞) satisfying p− = ess inf{p(x) : x ∈ E} > 1,
p+ = ess sup{p(x) : x ∈ E} < ∞.
The set P 0 (E) consists of all p(·) : E → (0, ∞) satisfying p− = ess inf{p(x) : x ∈ E} > 0,
p+ = ess sup{p(x) : x ∈ E} < ∞.
Definition 1.1 Let p(·) : E → [1, ∞) be a measurable function. (i) The Lebesgue space with variable exponent Lp(·) (E) is defined by Z |f (x)| p(x) Lp(·) (E) := f measurable : dx < ∞ for some constant η > 0 . η E p(·)
(ii) The space Lloc (E) is defined by p(·)
Lloc (E) := {f : f ∈ Lp(·) (F ) for all compact subsets F ⊂ E}. Lp(·) (E) is a Banach function space when it is equipped with the LuxemburgNakano norm Z |f (x)| p(x) dx 6 1 . kf kLp(·) (E) = inf η > 0 : η E Let f ∈ L1loc (Rn ). Then the Hardy-Littlewood maximal operator is defined by Z 1 M f (x) = sup |f (y)|dy, r>0 |Br (x)| Br (x) where Br (x) =
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