On Dirichlet type spaces and \(\alpha\)-Bloch spaces in the unit ball of \(\mathbb C^n\)
Let \(B\) be the unit ball of the complex \(n\)-space \(\mathbb C^n\). Given \(0<\alpha<\infty\), by the \(\alpha\)-Bloch space \(\beta^\alpha\) the authors mean the space of all holomorphic functions \(f\) on \(B\) such that \[ \sup_{z\in B}| \nabla f(z)| (1-| z| ^2)^\alpha <\infty. \] Also, given \(\tau\) real, by the Dirichlet type space \(D_\tau\) the authors mean the space of all holomorphic functions \(f=\sum_\alpha a_\alpha z^\alpha\) on \(B\) such that \(\sum_{\alpha}(| \alpha| +n)^\tau | a_\alpha| ^2\| z^\alpha\| ^2<\infty\) where the sum is taken over all multi-indices \(\alpha\) and \(\| z^\alpha\| \) is the \(H^2\) norm of the monomial \(z^\alpha\). These Dirichlet type spaces interpolate well-known function spaces such as the classical Dirichlet space (\(D_1\)), the Hardy space(\(D_0\)), and the Bergman space (\(D_{-1}\)). In this paper the authors study the inclusion relations between the spaces \(\beta^\alpha\) and \(D_\tau\). Main results are (i) \(\beta^\alpha\subset D_\tau\) for \(\tau <2-2\alpha\) and (ii) \(D_\tau\subset \beta^{1+{{n-\tau}\over 2}}\) for \(\tau<n+2\). Also, the authors construct examples of gap series to show that those inclusion relations are strict and sharp. Moreover, the authors prove the sharp inclusion relation \(D_n\subset BMOA\) which was conjectured by \textit{P. Y. Hu} and \textit{J. H. Shi} [Chin. Ann. Math., Ser. B 20, 369--376 (1999; Zbl 0941.32004)].
scientific article; zbMATH DE number 5000930