The Harmonic Bloch and Besov Spaces on the Real Unit Ball by an Oscillation. (16th March 2016) - Fu, Xi; Xu, Zhiyao; Liu, Xiaoyou
Abstract : Let B be the real unit ball in R n and f ∈ C N ( B ) . Given a multi-index m = ( m 1, …, m n ) of nonnegative integers with | m | = N, we set the quantity s u p x ∈ B, y ∈ E ( x, r ), x ≠ y ( 1 - | x | 2 ) α ( 1 - | y | 2 ) β | ∂ m f ( x ) - ∂ m f ( 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.