Double integral estimates for Besov type spaces and their applications

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.