Boundedness of area operators on Bergman spaces

TL;DR

This paper fully characterizes when area operators are bounded from Bergman spaces to Lebesgue spaces across all dimensions, extending previous partial results and solving open cases.

Contribution

It provides a complete characterization of boundedness for area operators on Bergman spaces in all complex dimensions, including new cases and extensions.

Findings

Complete boundedness criteria for area operators

Extension of results to higher dimensions

Resolution of previously open cases

Abstract

We completely characterize the boundedness of the area operators from the Bergman spaces $A^p_\alpha(\mathbb{B}_ n)$ to the Lebesgue spaces $L^q(\mathbb{S}_ n)$ for all $0<p,q<\infty$. For the case $n=1$, some partial results were previously obtained by Wu. Especially, in the case $q<p$ and $q<s$, we obtain the new characterizations for the area operators to be bounded. We solve the cases left open there and extend the results to $n$-complex dimension.