Besov-Lipschitz and mean Besov-Lipschitz spaces of holomorphic functions on the unit ball

TL;DR

This paper characterizes holomorphic mean Besov-Lipschitz and Besov-Lipschitz spaces on the unit ball, establishing equivalences between various norms and derivatives, and including polynomial approximation methods.

Contribution

It provides new characterizations and equivalences of these function spaces using derivatives, moduli of continuity, and polynomial approximation.

Findings

Different norms involving $L^p$-moduli of continuity are equivalent.

Characterizations involve radial, gradient, and tangential derivatives.

Polynomial approximation characterizations are established.

Abstract

We give several characterizations of holomorphic mean Besov-Lipschitz space on the unit ball in $\cn $ and appropriate Besov-Lipschitz space and prove the equivalences between them. Equivalent norms on the mean Besov-Lipschitz space involve different types of $L^p$-moduli of continuity, while in characterizations of Besov-Lipschitz space we use not only the radial derivative but also the gradient and the tangential derivatives. The characterization in terms of the best approximation by polynomials is also given.