A characterization of Bloch space and Besov space
The analytic Besov space \(B_p\), \(1<p<\infty\) on the unit disk \(D\) is defined to be the space of all analytic functions \(f\) on \(D\) such that \(| f'(z)| (1-| z| ^2)\) is \(L^p\)-integrable with respect to the invariant measure. A characterization with derivatives replaced by differences is also well known. That is, it is well known that an analytic function \(f\) is a member of \(B_p\) if and only if the difference function \(f(z)-f(w)\) is \(L^p\)-integrable on \(D\times D\) with respect to the weight given by the square of the absolute value of the Bergman kernel. In this paper the author obtains a new characterization, involving both differences and derivatives, that contains those two characterizations as special cases. A similar characterization that interpolates the two well-known characterizations for the Bloch space is also obtained. The key tool is a new characterization for the Bergman space that also interpolates the very definition and the derivative integrability characterization of the Bergman space. Recently, this new characterization for the Bergman space has been also proved on higher dimensional balls independently by \textit{M. Pavlović} and \textit{K. Zhu} [New characterizations of Bergman spaces (to appear)].
scientific article; zbMATH DE number 5073035
scientific article