Bergman spaces on the complement of a lattice

Abstract

We investigate Bergman spaces Bp (Ω), where Ω = C / (Z + iZ) and show that Bp = {0} for p ≥ 2 and {0} ≠ B q ⊂ Bp for 2/(n + 1) ≤ q < p < 2/n. Further, for each 0 < p < 2 there is a non-trivial f ∈ Bp tending to zero at infinity at any prescribed rate. We also give conditions on the Mittag-Leffler expansion of f necessary for f ∈ Bp.