Double integral estimates for Besov type spaces and their applications

TL;DR

This paper characterizes when certain integral conditions on analytic functions in the unit disk are equivalent, providing geometric insights into Besov-type spaces with weights and analyzing related Hankel operators.

Contribution

It offers a complete characterization of weighted integral conditions for Besov-type spaces and explores their geometric properties and operator boundedness.

Findings

Equivalent integral conditions for Besov spaces established

Geometric descriptions of functions in weighted Besov spaces provided

Boundedness and compactness criteria for Hankel operators derived

Abstract

For $0<p<\infty$, we give a complete description of nonnegative radial weight functions $\omega$ on the open unit disk $\mathbb{D}$ such that $$ \int_{\mathbb{D}} |f'(z)|^p (1-|z|^2)^{p-2}\omega(z)dA(z)<\infty $$ if and only if $$ \int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(\zeta)|^p}{|1-\overline{\zeta}z|^{4+\tau+\sigma}}(1-|z|^2)^{\tau}(1-|\zeta|^2)^{\sigma}\omega(\zeta)dA(z)A(\zeta)<\infty $$ for all analytic functions $f$ in $\mathbb{D}$, where $\tau$ and $\sigma$ are some real numbers. As applications, we give some geometric descriptions of functions in Besove type spaces $B_p(\omega)$ with doubling weights, and characterize the boundedness and compactness of Hankel type operators related to Besov type spaces with radial B\'ekoll\'e-Bonami weights. Some special cases of our results are new even for some standard weighted Besov spaces.