Double integral estimates for Besov type spaces and their applications

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 $${\int }_{\mathbb{D}}|f^{\prime }(z){|}^p(1-|z{|}^2{)}^{p-2}\omega (z)dA(z)<\infty$$ 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 Békollé-Bonami weights. Some special cases of our results are new even for some standard weighted Besov spaces.

V2, made some corrections on V1 and added some results