Boundedness of Sublinear Operators and Commutators on Generalized Morrey Spaces

Abstract

In this paper the authors study the boundedness for a large class of sublinear operators Tα, α ∈ [0, n) generated by Calderón-Zygmund operators (α = 0) and generated by Riesz potential operator (α > 0) on generalized Morrey spaces Mp,ϕ. As an application of the above result, the boundeness of the commutator of sublinear operators T b,α , α ∈ [0, n) on generalized Morrey spaces is also obtained. In the case b ∈ BM O and T b,α is a sublinear operator, we find the sufficient conditions on the pair (ϕ1, ϕ2) which ensures the boundedness of the operators T b,α , α ∈ [0, n) from one generalized Morrey space Mp,ϕ 1 to another Mq,ϕ 2 with 1/p − 1/q = α/n. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on (ϕ1, ϕ2), which do not assume any assumption on monotonicity of ϕ1, ϕ2 in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.Mathematics Subject Classification (2010). Primary 42B20, 42B25, 42B35.

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.…”

Generalized Morrey estimates for the gradient of divergence form parabolic operators with discontinuous coefficients

“…Let be a bounded (δ, 1)-Reifenberg flat domain verifying (19), and C and D be as above. Suppose that for any (y, τ ) ∈ C there exists ε ∈ (0, 1) such that (1) < ε and…”

“…These results allow to study the regularity of the solutions of various linear elliptic and parabolic boundary value problems in M p,ϕ (see [8,28,30]). A further development of the generalized Morrey spaces can be found in the works of Guliyev [15][16][17], see also [1,[18][19][20][21][22] and the references therein. Here we consider a supremum condition on the weight (14) which is optimal and ensure the boundedness of the Hardy-Littlewood maximal operator in M p,ϕ .…”

A characterization for fractional integral and its commutators in Orlicz and generalized Orlicz–Morrey spaces on spaces of homogeneous type

“…For the boundedness of I α on generalized Morrey spaces see [17,18,23,32] and references therein. The fractional integral in Orlicz spaces was studied in [7,24,28,33].…”