Area type inequalities and integral means of harmonic functions on the unit ball

Abstract

In this paper we investigate the relationship among the following integrals $${\int }_B\left|u\right(x\left){|}^{p-i}\right|\nabla u\left(x\right){|}^i(1-|x\left|{)}^{\alpha }dV\right(x),$$where $i\in \{0,1,2\}$, , $\alpha >0$, and where $u$ is an arbitrary harmonic function on the unit ball $B\subset {\mathbf{R}}^n$. Growth of the integral means of harmonic functions is also compared to the integral means of their gradient.