Kelvin-Möbius-Invariant Harmonic Function Spaces on the Real Unit Ball

Abstract

Abstract We define the Kelvin-Mobius transform of a function harmonic on the unit ball of R n and determine harmonic function spaces that are invariant under this transform. When n ≥ 3 , in the category of Banach spaces, the minimal Kelvin-Mobius-invariant space is the Bergman-Besov space b − ( 1 + n / 2 ) 1 and the maximal invariant space is the Bloch space b ( n − 2 ) / 2 ∞ . There exists a unique strictly Kelvin-Mobius-invariant Hilbert space, and it is the Bergman-Besov space b − 2 2 . There is a unique Kelvin-Mobius-invariant Hardy space.