Boundedness of a Class of Sublinear Operators and Their Commutators on Generalized Morrey Spaces

Abstract

The authors study the boundedness for a large class of sublinear operatorTgenerated by Calderón-Zygmund operator on generalized Morrey spacesMp,φ. As an application of this result, the boundedness of the commutator of sublinear operatorsTaon generalized Morrey spaces is obtained. In the casea∈BMO(ℝn),1<p<∞andTais a sublinear operator, we find the sufficient conditions on the pair (φ1,φ2) which ensures the boundedness of the operatorTafrom one generalized Morrey spaceMp,φ1to anotherMp,φ2. In all cases, the conditions for the boundedness ofTaare given in terms of Zygmund-type integral inequalities on (φ1,φ2), which do not assume any assumption on monotonicity ofφ1,φ2inr. Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator, and Bochner-Riesz operator.

Commutators of sublinear operators generated by Calderón-Zygmund operator on generalized weighted Morrey spaces

“…holds for all non-negative and non-increasing g on (0, ∞) if and only if Note that Theorem 3.1 is proved analogously to Theorem 4.3 in [15]. (1) The John-Nirenberg inequality: there are constants C 1 , C 2 > 0 such that for all…”

“…Note that Theorem 1.1 was proved in the case w ≡ 1 in [15] and in the case w ≡ 1 and ϕ(x, r) = ϕ 1 (x, r) = ϕ 2 (x, r) satisfying conditions (1.4) and (1.5) in [9].…”

Boundedness of Vector-Valued Intrinsic Square Functions in Morrey Type Spaces

“…For the boundedness of the Hardy-Littlewood maximal operator, the fractional integral operator and the Calderón-Zygmund singular integral operator on these spaces, we refer the reader to [1,2,13]. In [9], Mizuhara introduced the generalized Morrey spaces L p,Φ which was later extended and studied by many authors (see [4][5][6]8,11]). In [7], Komori and Shirai defined the weighted Morrey spaces L p,κ (w) which could be viewed as an extension of weighted Lebesgue spaces, and then discussed the boundedness of the above classical operators in Harmonic Analysis on these weighted spaces.…”

Vector-Valued Inequalities in the Morrey Type Spaces

“…For the boundedness of the Hardy-Littlewood maximal operator, the fractional integral operator, and the Calderón-Zygmund singular integral operator on these spaces, we refer the reader to [10][11][12]. In [13], Mizuhara introduced the generalized Morrey spaces ,Φ which were later extended and studied by many authors (see [14][15][16][17][18]). In [19], Komori and Shirai defined the weighted Morrey spaces , ( ) which could be viewed as an extension of weighted Lebesgue spaces and then studied the boundedness of the above classical operators in harmonic analysis on these weighted spaces.…”