Boundedness of Harmonic Conjugation on Weighted Bergman Spaces

TL;DR

The paper addresses when the harmonic conjugation operator is bounded on weighted harmonic Bergman spaces with Bekollé-Bonami weights, under a supplementary $p$-dependent assumption. It combines a Hardy–Littlewood–style estimate that controls $|f'(z)|$ by oscillations of the harmonic function $u$ with a novel good-λ argument to transfer these bounds to the norm in the weighted Bergman space, using a uniform delta-norm framework. The main contributions are (i) a pointwise HL-type bound for $|f'(z)|$ in terms of $u$, (ii) a dyadic Carleson-square and good-λ weighted inequality showing $ orm{B} leq K orm{D}$, and (iii) deduction that $ orm{f-f(0)}$ is controlled by $\norm{f'\delta_b}$, yielding boundedness of the harmonic conjugation operator on $a^p(w\,dA)$ under BB weights. This advances the theory by extending classical unweighted Bergman results to the BB-weighted setting and suggests broader applicability of the good-λ approach in harmonic analysis on Bergman spaces.

Abstract

We prove that if a weight is a Bekollé-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.

Theorems & Definitions (12)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Theorem 2.6
• proof
• Theorem 2.7