A canonical decomposition of automorphic forms which vanish on an invariant measurable subset
Let \(\Gamma <PSL(2;{\mathbb{R}})\) be a discrete subgroup and \(\Omega\) (\(\Gamma)\) its region of discontinuity. Consider a \(\Gamma\)-invariant closed subset \(\sigma\) of the extended real line and let D be the component of \(\Omega\) (\(\Gamma)\)-\(\sigma\) containing the upper half- plane. Suppose that E is a \(\Gamma\)-invariant subset of D and put \(V=D- E\). For any integer \(q\geq 2\) let \(L^ p\) (1\(\leq p\leq \infty)\) be the Banach space of all p-integrable (resp. bounded) measurable \(\Gamma\)- automorphic forms of weight -2q on D, and let \(A^ p\) be the closed subspace of all holomorphic functions in \(L^ p\). Put \[ L^ p(V)=\{f\in L^ p: f| E=0\}\quad and\quad A^ p| V=\{f\chi_ V: f\in A^ p\}, \] where \(\chi_ V\) is the indicator function of V, and let p' be such that \(1/p+1/p'=1\). The aim of the paper under review is to establish conditions on E which imply that \((A^ p)^{\perp}\cap L^{p'}(V)\) and \(A^{p'}| V\) are closed and complementary to each other in \(L^{p'}(V)\).
scientific article; zbMATH DE number 4023674
scientific article