Two-weight, weak type norm inequalities for a class of sublinear operators on weighted Morrey and amalgam spaces

TL;DR

This paper establishes two-weight weak type inequalities for a class of sublinear operators and their commutators on weighted Morrey and amalgam spaces, extending boundedness results under specific conditions.

Contribution

It provides new boundedness criteria for sublinear operators and their commutators on weighted Morrey and amalgam spaces, based on weak-type inequalities on weighted Lebesgue spaces.

Findings

Derived boundedness criteria for operators and commutators.

Established weak-type inequalities for several integral operators.

Extended results to weighted Morrey and amalgam spaces.

Abstract

Let $\mathcal T_\alpha~(0\leq\alpha<n)$ be a class of sublinear operators satisfying certain size conditions introduced by Soria and Weiss, and let $[b,\mathcal T_\alpha]~(0\leq\alpha<n)$ be the commutators generated by $\mathrm{BMO}(\mathbb R^n)$ functions and $\mathcal T_\alpha$. 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 criterions 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.