Boundedness of commutators generated by m-th Calder\'on-Zygmund type singular integrals and local Campanato functions on generalized local Morrey spaces

TL;DR

This paper investigates the boundedness properties of m-th Calderón-Zygmund type singular integrals and their commutators with local Campanato functions on generalized local Morrey spaces, extending understanding of these operators in harmonic analysis.

Contribution

It establishes the boundedness of these singular integrals and their commutators on generalized local Morrey spaces, a novel extension in the theory of harmonic analysis.

Findings

Boundedness of $T_m$ on generalized local Morrey spaces.

Boundedness of commutators with local Campanato functions.

Extension of boundedness results to product spaces.

Abstract

Let $T_m$ be the $m$-th Calder\'on-Zygmund type singular integral. In the paper, we consider the boundedness of $T_m$ on the generalized product local Morrey spaces $LM_{p_1,\varphi_1}^{\{x_0\}}\times LM_{p_2,\varphi_2}^{\{x_0\}}\times\dots\times LM_{p_m,\varphi_m}^{\{x_0\}}.$ And, the boundedness of the commutators of $T_m$ with local Campanato functions is obtained, also.