A remark on BMO and Bloch functions on the unit ball of \(\mathbb{R}^ n\)
Let \(B\) denote the unit ball in \(\mathbb{R}^ n\) \((n\geq 2)\). For a function \(f\) harmonic in \(B\), \(\| f\|_ \beta\) is defined by \[ \| f\|_ \beta= \sup_{x\in B} {\textstyle {1\over2}} (1-| x|^ 2) |\nabla f(x)|. \] It is shown that if \(f\) is harmonic in \(B\) then \[ \| f\|_{\text{BMO}}\leq \sqrt{2} \| f\|_ \beta, \qquad \| f\|_ \beta \leq (n+2)^{1/2} \| f\|_{\text{BMO}}. \] In the proof a simple variant of Green's formula is used.
scientific article; zbMATH DE number 558838