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On integral equations related to weighted Toepitz operators

For weighted Toeplitz operators $\T^N_ϕ$ defined on spaces of holomorphic functions in the unit ball, we derive regularity properties of the solutions $f$ to the integral equation $\T^N_ϕ(f)=h$ in terms of the regularity of the symbol $ϕ$ 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.