In this talk, we discuss various aspects of weighted kernel functions on planar domains. We focus on two key kernels, namely, the weighted Bergman kernel and the weighted Szegő kernel.

For a planar domain $D \subset \mathbb C$ and an admissible weight function $\mu$ on it, we discuss some aspects of the corresponding weighted Bergman kernel $K_{D, \mu}$. First, we see a precise relation between $K_{D, \mu}$ and the classical Bergman kernel $K_D$ near a smooth boundary point of $D$. Second, the weighted kernel $K_{D, \mu}$ gives rise to weighted metrics in the same way as the classical Bergman kernel does. Motivated by work of Mok, Ng, Chan–Yuan and Chan–Xiao–Yuan among others, we talk about the nature of holomorphic isometries from the disc $\mathbb D \subset \mathbb C$ with respect to the weighted Bergman metrics arising from weights of the form $\mu = K_{\mathbb D}^{-d}$ for some integer $d \geq 0$. Specific examples that we discuss in detail include those in which the isometry takes values in $\mathbb D^n$ and $\mathbb D \times \mathbb B^n$ where each factor admits a weighted Bergman metric as above for possibly different non-negative integers $d$. Finally, we also present the case of isometries between polydisks in possibly different dimensions, in which each factor has a different weighted Bergman metric as above.

In the next part of the talk, we discuss properties of weighted Szegő and Garabedian kernels on planar domains. Motivated by the unweighted case as explained in Bell’s work, the starting point is a weighted Kerzman–Stein formula that yields boundary smoothness of the weighted Szegő kernel. This provides information on the dependence of the weighted Szegő kernel as a function of the weight. When the weights are close to the constant function $1$ (which corresponds to the unweighted case), we show that some properties of the unweighted Szegő kernel propagate to the weighted Szegő kernel as well. Finally, we show that the reduced Bergman kernel and higher order reduced Bergman kernels can be written as a rational combination of three unweighted Szegő kernels and their conjugates, thereby extending Bell’s list of kernel functions that are made up of simpler building blocks that involve the Szegő kernel.

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Last updated: 29 Feb 2024