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

Abstract

Let ${\mathcal{T}}_{\alpha }(0\le \alpha <n)$ be a class of sublinear operators satisfying certain size conditions introduced by Soria and Weiss, and let $[b,{\mathcal{T}}_{\alpha }](0\le \alpha \le n)$ be the commutators generated by BMO(${ℝ}^n$) 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 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.