On a BMO-property for subharmonic functions
A compact set \(K\subset \mathbb{R}^2\) is called (Ahlfors) \(d\)-regular, if there exists \(a>0\) such that for any \(x\in K\) and \(0<t\leq \text{diam}(K)\) \[ \mathcal{H}^d(K_{x,t})\leq at^d \] where \(K_{x,t}=\{z\in \mathbb{C}: |x-z|<t\}\cap K\) and \(\mathcal{H}^d(\omega)\) stands for the \(d\)-Hausdorff measure of \(\omega\). If also \(bt^d \leq \mathcal{H}^d(K_{x,t})\), then the set \(K\) is said to be a \(d\)-set. The author proves that, for any subharmonic function defined on an open neighbourhood of \(K\), its restriction \(f|_{K}\) belongs to \(\text{BMO}(K, \mathcal{H}^d)\).
scientific article; zbMATH DE number 1921139