Boundedness of $\theta$-type Calder\'on--Zygmund operators and commutators in the generalized weighted Morrey spaces

TL;DR

This paper introduces new Morrey-type spaces and investigates the boundedness of $ heta$-type Calderón--Zygmund operators and their commutators within these spaces, establishing various strong, weak, and endpoint estimates.

Contribution

It develops a new class of Morrey-type spaces and analyzes the boundedness of Calderón--Zygmund operators and their commutators in these spaces, including two-weight inequalities.

Findings

Strong and weak type estimates for $T_ heta$ in new Morrey spaces.

Boundedness and endpoint estimates for commutators $[b,T_ heta]$.

Partial results on two-weight, weak type inequalities.

Abstract

In this paper, we first introduce some new Morrey type spaces containing generalized Morrey space and weighted Morrey space as special cases. Then we discuss the strong type and weak type estimates for a class of Calder\'on--Zygmund type operators $T_\theta$ in these new Morrey type spaces. Furthermore, the strong type estimate and endpoint estimate of commutators $[b,T_{\theta}]$ formed by $b$ and $T_{\theta}$ are established. Also we study related problems about two-weight, weak type inequalities for $T_{\theta}$ and $[b,T_{\theta}]$ in the Morrey type spaces and give partial results.