On the behavior near a torus of functions holomorphic in the ball
Let B be the open unit ball in \({\mathbb{C}}^ n\). Let \(d\mu (r,\theta)=(1- r)^{\alpha}drd\theta\), \(\alpha >-1\), in polar coordinates. Denote by \(A^ P_{\alpha}\) the space of all holomorphic function in \(L^ P(d\mu)\). The main results of the paper read as follows. Let \(\pi:{\mathbb{C}}^ n\to {\mathbb{C}}\) be given by \(\pi (z_ 1,...,z_ n)=n^{n/2}\prod^{n}_{j=1}z_ j.\) Denote by \(H^ 2(B)\) the Hardy space over B and by BMO(B) the space of all functions of bounded mean oscillation in B. (a) If \(g\in A^ P_{(n-3)/2}\), then \(g\circ \pi \in H^ p(B)\); (b) if g is a Bloch function in the open unit disc U then \(g\circ \pi \in BMO(B)\). - As a corollary of the above (b) it is shown that there is a holomorphic function in BMO(B) which has a finite radial limit at no point of \(T_ n=\{z\in {\mathbb{C}}^ n,\) \(| z_ j| =n^{-1/2},\) \(j=1,...,n\}.\)
scientific article; zbMATH DE number 3881034