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

Abstract EN

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.