Two-Weight, Weak-Type Norm Inequalities for a Class of Sublinear Operators on Weighted Morrey and Amalgam Spaces

Hua Wang

doi:10.1155/2020/3673921

Abstract

Let Tα0≤α<n be a class of sublinear operators satisfying certain size conditions introduced by Soria and Weiss, and let b,Tα0≤α<n be the commutators generated by BMORn functions and Tα. This paper is concerned with two-weight, weak-type norm estimates for these sublinear operators and their commutators on the weighted Morrey and amalgam spaces. Some boundedness criteria for such operators are given, under the assumptions that weak-type norm inequalities on weighted Lebesgue spaces are satisfied. As applications of our main results, we can obtain the weak-type norm inequalities for several integral operators as well as the corresponding commutators in the framework of weighted Morrey and amalgam spaces.

Highlights

Let b be a locally integrable function on Rn; suppose that the commutator operator 1⁄2b, T stands for a linear or a sublinear operator, which satisfies that for any f ∈ L1ðRnÞ with compact support and x ∉ supp f, j1⁄2b, T
The main purpose of this paper is to investigate the two-weight, weak-type norm inequalities in the setting of weighted Morrey and amalgam spaces
We conclude the proof of Theorem 12 by taking the supremum over all l > 0

Summary

Introduction and Main Results

Let b be a locally integrable function on Rn; suppose that the commutator operator 1⁄2b, T stands for a linear or a sublinear operator, which satisfies that for any f ∈ L1ðRnÞ with compact support and x ∉ supp f , j1⁄2b,. Let 1 < p < ∞, 0 ≤ κ < 1, and w be a weight on Rn. We define the weighted weak Morrey space WLp,κðwÞ as the set of all measurable functions f satisfying kf kW Lp,κ ðwÞ. The main purpose of this paper is to investigate the two-weight, weak-type norm inequalities in the setting of weighted Morrey and amalgam spaces. Given a pair of weights ðw, νÞ, assume that for some r > 1 and for all cubes Q in Rn, ð26Þ where AðtÞ = tp′1⁄2log ðe + tÞp′. We are able to apply our main theorems to these classical integral operators

Notations and Definitions

Proofs of Theorems 5 and 6

Proofs of Theorems 7 and 8

Proofs of Theorems 9 and 10

Proofs of Theorems 11 and 12