Bergman spaces on the complement of a lattice (Q1423835)

Let \(\Gamma = \mathbb{Z} + i\mathbb{Z}\) and put \(\Omega = \mathbb{C} \setminus\Gamma\). The present paper deals with the spaces \(B^p(\Omega)\) of analytic functions \(f\) on \(\Omega\) which satisfy \[ \| f\|_p=\left(\iint_\Omega | f(x+iy)|^p\,dx\,dy\right)^{1/p}<\infty. \] The authors study inclusion properties for the Bergman spaces \(B^p(\Omega)\) and Mittag-Leffler expansions of the elements \(f\in B^p(\Omega)\). For example, it is shown that \(B^p(\Omega) = \{0\}\) whenever \(p\geq 2\). If, for some \(n = 1, 2,\dots,2/(n + 1)\leq q<p< 2/n\) then \(B^q(\Omega)\subset B^p(\Omega)\). Moreover, if \(1 \leq p < 2\) and \(f\in B^p(\Omega)\), then \[ f(z)=\sum_{\omega\in\Gamma}\;\frac{a_\omega}{z-\omega}\text{ for some }a_\omega,\text{ where }\sum_{\omega\in\Gamma}| a_\omega|^p <\infty. \] This expansion of \(f\) converges locally uniformly if one considers a special order on \(\Gamma\) called the ``spiral ordering'' in the article. Finally, the authors study growth conditions for \(f\in B^p(\Omega)\). They note that all rational \(f\in B^p(\Omega)\) satisfy \[ f(z)=O(| z| -2/p),\quad | z|\to \infty,\text{ where }z\in\Omega_\varepsilon :=\{z\in\mathbb{C}:d(z,\Gamma)>\varepsilon\}. \] On the other hand, they prove that the implication \[ f\in B^p(\Omega)\Rightarrow f(z)=O(| z|^{-\alpha}),\quad | z|\to\infty,\;z\in\Omega_\varepsilon, \] does not hold for any \(0 < p< 2\), \(\alpha > 0\), \(0<\varepsilon < 1/\sqrt 2\).