Kelvin-Möbius-invariant harmonic function spaces on the real unit ball

We define the Kelvin-Möbius transform of a function harmonic on the unit ball of Rn and determine harmonic function spaces that are invariant under this transform. When n ≥ 3, in the category of Banach spaces, the minimal Kelvin-Möbius-invariant space is the Bergman-Besov space b1−(1+n/2) and the maximal invariant space is the Bloch space b∞(n−2)/2. There exists a unique strictly Kelvin-Möbius-invariant Hilbert space, and it is the Bergman-Besov space b2−2. There is a unique Kelvin-Möbius invariant Hardy space.