On Integral Equations Related to Weighted Toeplitz Operators

For weighted Toeplitz operators T^N_j{{\mathcal T}^N_\varphi} defined on spaces of holomorphic functions in the unit ball, we derive regularity properties of the solutions f to the equation T^N_j(f)=h{{\mathcal T}^N_\varphi(f)=h} in terms of the regularity of the symbol φ and the data h. As an application, we deduce that if f\not o 0{f\not\equiv0} is a function in the Hardy space H ¹ such that its argument `(f)]/f{\bar f/f} is in a Lipschitz space on the unit sphere \mathbb S{{\mathbb S}}, then f is also in the same Lipschitz space, extending a result of Dyakonov to several complex variables.