The Harmonic Bloch and Besov Spaces on the Real Unit Ball by an Oscillation

Abstract

Let B be the real unit ball in Rn and f∈CN(B). Given a multi-index m=(m1,…,mn) of nonnegative integers with |m|=N, we set the quantity supx∈B,y∈E(x,r),x≠y(1-|x|2)α(1-|y|2)β|∂mf(x)-∂mf(y)|/|x-y|γ[x,y]1-γ,  x≠y, where 0≤γ≤1 and α+β=N+1. In terms of it, we characterize harmonic Bloch and Besov spaces on the real unit ball. This generalizes the main results of Yoneda, 2002, into real harmonic setting