Two-weight, weak type norm inequalities for fractional integral operators and commutators on weighted Morrey and amalgam spaces

TL;DR

This paper establishes two-weight, weak type norm inequalities for fractional integral operators and their commutators on weighted Morrey and amalgam spaces, extending known results from Lebesgue spaces.

Contribution

It introduces new weak-type norm inequalities for fractional integrals and commutators on weighted Morrey and amalgam spaces using $A_p$-type conditions.

Findings

Weak-type inequalities for fractional integrals established.

Results extend to commutators with symbol functions.

Includes estimates for extreme cases on weighted spaces.

Abstract

Let $0<\gamma<n$ and $I_\gamma$ be the fractional integral operator of order $\gamma$, $I_{\gamma}f(x)=\int_{\mathbb R^n}|x-y|^{\gamma-n}f(y)\,dy$, and let $[b,I_\gamma]$ be the linear commutator generated by a symbol function $b$ and $I_\gamma$, $[b,I_{\gamma}]f(x)=b(x)\cdot I_{\gamma}f(x)-I_\gamma(bf)(x)$. This paper is concerned with two-weight, weak type norm estimates for such operators on the weighted Morrey and amalgam spaces. Based on weak-type norm inequalities on weighted Lebesgue spaces and certain $A_p$-type conditions on pairs of weights, we can establish the weak-type norm inequalities for fractional integral operator $I_{\gamma}$ as well as the corresponding commutator in the framework of weighted Morrey and amalgam spaces. Furthermore, some estimates for the extreme case are also obtained on these weighted spaces.