Harmonic Bergman spaces on the complement of a lattice

Abstract

We investigate harmonic Bergman spaces bp = bp(Ω), 0 < p < ∞, where Ω = ℝn \ ℤn and prove that bq ⊂ bp for n/(k + 1) ≤ q < p < n/k. In the planar case we prove that bp is non empty for all 0 < p < ∞. Further, for each 0 < p < ∞ there is a non-trivial f ∈ bp tending to zero at infinity at any prescribed rate.