Weighted Bergman projections on the Hartogs triangle: exponential decay (Q502163)

Let \(\Omega\subset\mathbb C^n\) be a bounded domain and let \(\mu\) be a nonnegative weight function on \(\Omega\) such that \(L^2_a(\Omega,\mu)\), the space of square integrable holomorphic functions, is a closed subspace of \(L^2(\Omega,\mu)\), the space of square integrable functions with respect to \(\mu(z)dV(z)\), where \(dV(z)\) stands for the Lebesgue measure. Denote by \(B^{\mu}_{\Omega}\) the weighted Bergman projection operator \(L^2(\Omega,\mu)\longrightarrow L_a^2(\Omega,\mu)\). In the paper under review the weighted Bergman projection on the Hartogs triangle \[ \mathbb H:=\big\{(z_1,z_2)\in\mathbb C^2:|z_2|<|z_1|<1\big\} \] is studied. The authors present the following results: 1. Let \(\nu(x):=\exp(-1/x)\), \(\delta(z_1,z_2):=|z_1|\) and \(\mu(z):=\nu(\delta(z))\). Then (a) the weighted Bergman projection \(B^{\mu}_{\mathbb H}\) is bounded on \(L^p(\mathbb H,\mu)\) if and only if \(p=2\); (b) \(\delta(z)^{\tau}B^{\mu}_{\mathbb H}(z,z)\), where \(\tau>0\), is unbounded as \(z\) approaches the origin inside any cone \(V_{\gamma}:=\{(z_1,z_2)\in\mathbb C^2:\gamma|z_2|<|z_1|\}\), where \(\gamma>1\). 2. The Bergman projection \(B_{\mathbb H_{\infty}}\) is bounded on \(L^p(\mathbb H_{\infty})\) if and only if \(p=2\), where \[ \mathbb H_{\infty}:=\big\{(z_1,z_2)\in\mathbb C^2:|z_2|<\exp(-1/|z_1|)<1\big\}. \]