On integral equations related to weighted Toepitz operators

TL;DR

This paper investigates the regularity of solutions to integral equations involving weighted Toeplitz operators on holomorphic function spaces, extending known results to several complex variables and linking function argument regularity to Lipschitz spaces.

Contribution

It derives regularity properties of solutions to weighted Toeplitz integral equations and extends Dyakonov's result to functions in several complex variables.

Findings

Solutions' regularity depends on the symbol and data regularity.

If a Hardy space function's argument is Lipschitz, the function itself is Lipschitz.

Extension of Dyakonov's result to multiple complex variables.

Abstract

For weighted Toeplitz operators $\T^N_\phi$ defined on spaces of holomorphic functions in the unit ball, we derive regularity properties of the solutions $f$ to the integral equation $\T^N_\phi(f)=h$ in terms of the regularity of the symbol $\phi$ and the data $h$. As an application, we deduce that if $f\not\equiv0$ is a function in the Hardy space $H^1$ such that its argument $\bar f/f$ is in a Lipschitz space on the unit sphere $\bB$, then $f$ is also in the same Lipschitz space, extending a result of K. Dyakonov to several complex variables.