Integral means of the Poisson integral of a discrete measure
Let \(u\) denote a harmonic function in the open unit disc and put \[ M_ p (u,r)= \Biggl\{ {1\over {2\pi}} \int_ 0^{2\pi} | u(re^{i\theta})|^ p d\theta \Biggr\}^{1/p}, \qquad D= \partial/\partial \theta. \] Take \(0<p<1\) and integer \(n> (1-p)/p\). Then the following conditions (A) and (B) are shown to be equivalent: (A) \(M_ p (D^{n-1} u,r)= O(1-r)^{(1- np)/p}\), \(r\to 1\) and \(\lim_{r\to 1} u(re^{i\theta})=0\) for almost all \(\theta\). (B) \(u\) is the Poisson integral of a discrete measure \(\lambda\) on the unit circle such that \(\sum| \lambda\{w\}|^ p <\infty\).
scientific article; zbMATH DE number 635920
scientific article