An off-diagonal estimate of Bergman kernels
Let \(\mathbb D\) denote the unit disk in \(\mathbb C\) with area measure \(d\Sigma\). For a nonnegative weight \(\omega\) on \(\mathbb D\), \(L^{2}(\mathbb D,\omega)\) denotes the space of complex-valued functions on \(\mathbb D\) that are square integrable with respect to the measure \(\omega d\Sigma\) with norm \(\|f\|_{\omega}\), and \(A^2(\mathbb D,\omega)\) the closure of the analytic polynomials in \(L^2(\mathbb D,\omega)\). The space \(A^2(\mathbb D,\omega)\) is a weighted Bergman space with reproducing kernel \(K_{\omega}\) provided \(|p(z)|\leq C(X)\|p\|_{\omega}\) (\(z\in X\)) for all polynomials \(p\) and compact subsets \(X\) of \(\mathbb D\). In the paper the author proves that if \(\omega\) is logarithmically subharmonic on \(\mathbb D\) and reproducing for the origin, then the kernel \(K_{\omega}\) satisfies the following inequalities: for all \((z,\zeta)\in\mathbb D\times\mathbb D\), \[ \frac 12K_{\omega}(\zeta,\zeta) \frac{(1-|\zeta|^2)^2}{|1-z\overline\zeta|^2} \leq |K_{\omega}(z,\zeta)|\leq \frac 2{|1-z\overline \zeta|^2}. \]
scientific article; zbMATH DE number 1463870
scientific article