Mean Lipschitz conditions and growth of area integral means of functions in Bergman spaces with an admissible Békollé weight

Abstract

Galanopoulos et al. proved that the mean Lipschitz condition for $f$ in the classical Bergman space is characterized by the growth of the area integral mean of its derivative as well as by the growth of the norm of the difference between $f$ and the dilated function of $f$. We prove that functions in the weighted Bergman space with admissible Békollé weights also have the same property. Furthermore we investigate the Bloch and Zygmund-type spaces for admissible weight.