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

摘要

In this paper we investigate the relationship among the following integrals ∫_B |u(x)|~(p-i)|▽u(x)|~i(1-|x|~α dV(x), where i ∈ {0, 1, 2}, 1 < p < ∞, α > 0, and where u is an arbitrary harmonic function on the unit ball B is contained in R~n. Growth of the integral means of harmonic functions is also compared to the integral means of their gradient.