Endpoint Estimates for Bergman Commutators and New Characterizations of the Bloch Space and $H^\infty$

TL;DR

This paper establishes a distributional inequality for Bergman commutators involving Bloch functions, characterizes Bloch space membership, and explores the weak-type behavior of these operators, extending classical harmonic analysis results to the Bergman setting.

Contribution

It introduces a Bergman space analogue of a Calderón-Zygmund inequality, characterizes Bloch functions via this inequality, and analyzes the weak-type properties of Bergman commutators.

Findings

The inequality characterizes Bloch space membership.

The estimate is sharp, with counterexamples for weak-type (1,1).

Analytic symbols yield weak-type (1,1) if and only if they are bounded.

Abstract

We prove an $\LlogL $-type distributional inequality for the commutator of the Bergman projection with a conjugate Bloch symbol function on the unit ball. Such an inequality can be seen as a Bergman version of a result due to C. P\'{e}rez for real-variable Calder\'{o}n-Zygmund operators and BMO functions. We also prove that this inequality characterizes membership of analytic functions in the Bloch space and is further equivalent to a kind of modified restricted weak-type estimate, where one only tests over characteristic functions of sets comparable to Bergman balls. We also show our estimate is sharp in the sense that there exists a Bloch function $b$ so that the commutator $[\bar{b},P]$ is not weak-type $(1,1)$, and prove $[\bar{b},P]$ with $b$ analytic is weak-type $(1,1)$ if and only if $b \in H^\infty$.