An integral representation for Besov and Lipschitz spaces

TL;DR

The paper extends integral representation formulas to a broad class of Besov and Lipschitz spaces, including Bergman and Fock spaces, generalizing known results for the analytic Besov space $B_1$.

Contribution

It provides new integral representations for Besov spaces $B_p$ with $0<p extless=1$ and Lipschitz spaces $\Lambda_t$ with $t>1$, also applicable to Bergman and Fock spaces.

Findings

Derived integral representations for $B_p$ spaces with $0<p extless=1$.

Extended integral formulas to Lipschitz spaces $\Lambda_t$ with $t>1$.

Included representations for Bergman and Fock spaces.

Abstract

It is well known that functions in the analytic Besov space $B_1$ on the unit disk $\D$ admits an integral representation $$f(z)=\ind\frac{z-w}{1-z\bar w}\,d\mu(w),$$ where $\mu$ is a complex Borel measure with $|\mu|(\D)<\infty$. We generalize this result to all Besov spaces $B_p$ with $0<p\le1$ and all Lipschitz spaces $\Lambda_t$ with $t>1$. We also obtain a version for Bergman and Fock spaces.