On the convergence sets of operator sequences on spaces of homogeneous type

TL;DR

This paper characterizes the sets where sequences of operators on spaces of homogeneous type converge or diverge, providing a comprehensive understanding of their behavior for functions in various L^p spaces.

Contribution

It offers a complete characterization of convergence and divergence sets for operator sequences on spaces of homogeneous type, extending classical analysis results.

Findings

Provides necessary and sufficient conditions for convergence sets

Characterizes divergence sets of operator sequences

Applies results to specific classical analysis operators

Abstract

We consider sequences of operators $U_n:L^1(X)\to M(X)$, where $X$ is a space of homogeneous type. Under certain conditions on the operators $U_n$ we give a complete characterization of convergence (divergence) sets of functional sequences $U_n(f)$, where $f\in L^p(X)$, $1\le p\le \infty$. The results are applied to characterize convergence sets of some specific operator sequences in classical analysis.