Boundedness of Harmonic Conjugation on Weighted Bergman Spaces

TL;DR

This paper establishes conditions under which harmonic conjugation is bounded on weighted Bergman spaces, expanding understanding of operator boundedness in complex analysis with weighted function spaces.

Contribution

It introduces new boundedness criteria for harmonic conjugation on weighted Bergman spaces involving Bekollé-Bonami weights and a novel good lambda inequality approach.

Findings

Boundedness of harmonic conjugation under specific weight conditions

Introduction of a new good lambda inequality technique

Extension of boundedness results to broader weighted spaces

Abstract

We prove that if a weight is a Bekoll\'{e}-Bonami weight for some $q$ and it satisfies another simple condition that depends on $0 < p < \infty$, then the operator taking a function to its harmonic conjugate is bounded on the harmonic Bergman space $a^p$. One part of our results uses a certain special type of good lambda inequality.