Abstract
LetBbe the real unit ball inRnandf∈CN(B). Given a multi-indexm=(m1,…,mn)of nonnegative integers with|m|=N, we set the quantitysupx∈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,where0≤γ≤1andα+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.
Highlights
IntroductionWe denote the class of all harmonic functions on the unit ball by H(B)
Let B be the real unit ball in Rn with n ≥ 2, where dV is the normalized volume measure on B and dσ is the normalized surface measure on the unit sphere S = ∂B
For each α > 0, the harmonic α-Bloch space Bα consists of all functions f ∈ H(B) such that
Summary
We denote the class of all harmonic functions on the unit ball by H(B). For each α > 0, the harmonic α-Bloch space Bα consists of all functions f ∈ H(B) such that. In [2], Yoneda characterized harmonic Bloch and Besov spaces in D in terms of M as follows. Let f be a complex-value harmonic function in D. See [3,4,5] for various characterizations of the Bloch, little Bloch, and Besov spaces in the unit ball of Cn. The main purpose of this paper is to give some characterizations for the spaces Bα, Bα0 , and Bp in the real unit ball along Yoneda’s direction. Throughout this paper, constants are denoted by C; they are positive and may differ from one occurrence to the other. The notation A ≍ B means there is a positive constant C such that B/C ≤ A ≤ CB
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