Double integral estimates for Besov type spaces and their applications

For $0<p<\infty$, we give a complete description of nonnegative radial weight functions $ω$ on the open unit disk $\mathbb{D}$ such that $\int_{\mathbb{D}} |f'(z)|^p (1-|z|^2)^{p-2}ω(z)dA(z)<\infty$ if and only if $\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(ζ)|^p}{|1-\overlineζz|^{4+τ+σ}}(1-|z|^2)^τ(1-|ζ|^2)^σω(ζ)dA(z)A(ζ)<\infty$ for all analytic functions $f$ in $\mathbb{D}$, where $τ$ and $σ$ are some real numbers. As applications, we give some geometric descriptions of functions in Besove type spaces $B_p(ω)$ 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.

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