Harmonic Bergman Spaces on the Complement of a Lattice

Abejela Shkheam, Ali Abaob, Miloš Arsenović

We investigate harmonic Bergman spaces $b^p=b^p(\Omega)$, $0<p<\infty$, where $\Omega=\Bbb R^n\backslash\Bbb Z^n$ and prove that $b^q\subset b^p$, for $n/(k+1)\leq q<p<n/k$. In the planar case we prove that $b^p$ is non empty for all $0<p<\infty$. Further, for each $0<p<\infty$ there is a non-trivial $f\in B^p$ tending to zero at infinity at any prescribed rate.