Parabolic sublinear operators with rough kernel generated by parabolic Calder\'on-Zygmund operators and parabolic local Campanato space estimates for their commutators on the parabolic generalized local Morrey spaces

TL;DR

This paper introduces new parabolic generalized local Morrey spaces, proves boundedness of parabolic rough operators and estimates for their commutators, with applications to specific operators like the Marcinkiewicz operator.

Contribution

It develops a framework for analyzing parabolic rough operators and their commutators on newly defined Morrey spaces, extending previous results.

Findings

Boundedness of parabolic rough operators on generalized local Morrey spaces

Estimates for commutators in parabolic local Campanato spaces

Application to parabolic Marcinkiewicz operator

Abstract

In this paper, the author introduces parabolic generalized local Morrey spaces and gets the boundedness of a large class of parabolic rough operators on them. The author also establihes the parabolic local Campanato space estimates for their commutators on parabolic generalized local Morrey spaces. As its special cases, the corresponding results of parabolic sublinear operators with rough kernel and their commutators can be deduced, respec- tively. At last, parabolic Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.