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 ∫B|u(x)|p-i|∇u(x)|i(1 - |x| ) α, where i ∈ {0,1,2}, 1 < p < ∞, α >0, and where u is an arbitrary harmonic function on the unit ball B ⊂ Rn. Growth of the integral means of harmonic functions is also compared to the integral means of their gradient.