Moment vanishing properties of harmonic Bergman functions (Q1877786)

Let \(\mathbb{H}\) denote the halfspace \(\mathbb{R}^n\times (0,\infty)\). If \(\delta\in \mathbb{R}\) and \(0< p<\infty\), then \(b^p_\delta\) denotes the weighted harmonic Bergman space consisting of all harmonic functions \(u\) on \(\mathbb{H}\) such that the function \((x,t)\mapsto| u(x,t)|^p t^\delta\) is Lebesgue integrable on \(\mathbb{H}\). Theorem 1.1: Suppose \(N\) is a nonnegative integer and \(\mu\) is a complex Borel measure on \(\mathbb{R}^n\) such that \(| x|^N\in L^1(|\mu|)\). If \(\delta>-n-1\) and the Poisson integral of \(\mu\) in \(\mathbb{H}\) belongs to \(b^p_\delta\) for some \(p\in (0\), \((n+\delta+1)/(n+ N)]\), then \(\int_{\mathbb{R}^n} x^\alpha d\,\mu(x)= 0\) for all multi-indices \(\alpha\) satisfying \(|\alpha|\leq N\). The paper also presents related weighted norm inequalities for Poisson integrals, and some applications of these results.