Abstract
The aim of this paper is to get the boundedness of a class of sublinear operators with rough kernels on weighted Morrey spaces under generic size conditions, which are satisfied by most of the operators in classical harmonic analysis. Applications to the corresponding commutators formed by certain operators and BMO functions are also obtained.
Highlights
Introduction and Main ResultsGiven a function Ω over the unit sphere Sn−1 of Rn (n ≥ 2) equipped with the normalized Lebesgue measure dσ and x = x/|x|, a Calderon-Zygmund singular integral operator with rough kernel was given by Ω (x − y) TΩf (x) = p.v. ∫ Rnx − yn f (y) dy (1)and a related maximal operator MΩf sup r>0 1 rn
The topic of this paper is intended as an attempt to study the boundedness of sublinear operators with rough kernels which satisfy (7) and (8) on weighted Morrey spaces
Proofs of Theorems 1 and 3 depend heavily on some properties of Ap weights, which can be found in any papers or any books dealing with weighted boundedness for operators in harmonic analysis, such as [1]
Summary
When Ω satisfies some size conditions, the kernel of the operator TΩ has no regularity, and so the operator TΩ is called rough singular integral operator. In [9], Hu et al considered some more general sublinear operators with rough kernels which satisfy. Inequality (5) is satisfied by many operators with rough kernels in classical harmonic analysis, such as TΩ (see [11]) and the oscillatory singular integral operator. Inspired by the works of [6, 13], in this paper, we consider some sublinear operators under some size conditions (the following (7) and (8)) which are more general than (5):. The topic of this paper is intended as an attempt to study the boundedness of sublinear operators with rough kernels which satisfy (7) and (8) on weighted Morrey spaces.
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