Mapping properties of weighted Bergman projection operators on Reinhardt domains (Q2809202)

Let \(\Omega\) be a bounded domain in \(\mathbb C^n\), let \(\lambda\) be a positive continuous function on \(\Omega\), and let \(dV(z)\) denote the standard Lebesgue measure on \(\Omega\). The space of square integrable (resp.~square integrable holomorphic) functions on \(\Omega\) with respect to the measure \(\lambda(z)dV(z)\) is denoted by \(L^2(\Omega,\lambda)\) (resp.~\(L^2_a(\Omega,\lambda)\)). Recall that the weighted Bergman projection operator \(\boldsymbol{B}^{\lambda}_{\Omega}:L^2(\Omega,\lambda)\longrightarrow L^2_a(\Omega,\lambda)\) is of the form NEWLINE\[NEWLINE \boldsymbol{B}^{\lambda}_{\Omega}f(z)=\int_{\Omega}B^{\lambda}_{\Omega}(z,w)f(w)\lambda(w)dV(w), NEWLINE\]NEWLINE where \(B^{\lambda}_{\Omega}\) is the weighted Bergman kernel.NEWLINENEWLINEFor a given pair \(\Omega\) and \(\lambda\) a natural question arises: for which \(p\in(1,+\infty)\) is the operator \(\boldsymbol{B}^{\lambda}_{\Omega}\) bounded on \(L^p(\Omega,\lambda)\)? One of the well-studied settings is the case when \(\Omega\) is the unit disc in the complex plane.NEWLINENEWLINEIn the paper under review the authors study the higher-dimensional case. They present the following results.NEWLINENEWLINE1. Let \(\Omega\) be a smooth bounded complete Reinhardt domain in \(\mathbb C^n\) and let \(\rho\) be a smooth multi-radial defining function for \(\Omega\). If \(\lambda=\exp(1/\rho)\), then \(\boldsymbol{B}^{\lambda}_{\Omega}\) is bounded on \(L^p(\Omega,\lambda)\) if and only if \(p=2\).NEWLINENEWLINE2. Let \(\mathbb B^n\) denote the unit Euclidean ball in \(\mathbb C^n\) and let \(\mu(z)=\exp(1/(\|z\|^2-1))\). Then, for any \(k\in\mathbb N\), the weighted Bergman projection \(\boldsymbol{B}^{\mu}_{\mathbb B^n}\) is bounded on the weighted \(L^2\)-Sobolev space \(W^k(\mathbb B^n,\mu)\) with the norm NEWLINE\[NEWLINE \|f\|^2_{k,\mu}=\sum_{|\beta+\gamma|\leq k}\int_{\mathbb B^n}\left|\frac{\partial^{\beta+\gamma}}{\partial\bar z^{\beta}\partial z^{\gamma}}f(z)\right|^2\mu(z)dV(z). NEWLINE\]