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|)^αdV(x),
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.