Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Note on Kernel Functions of Dirichlet Spaces

Sahil Gehlawat, Aakanksha Jain, Amar Deep Sarkar

TL;DR

The paper studies how the $n$-th order weighted kernel functions $M_{\Omega,\mu,n}$, associated with Dirichlet spaces on planar domains, vary with the base point and the weight. It proves that $M_{\Omega,\mu,n}$ is $C^{\infty}$ off a thin set $N_{\Omega}(\mu)$, with explicit integral relations to the corresponding reduced Bergman kernels $\tilde{K}_{\Omega,\mu,n}$, and obtains a Ramadanov-type convergence theorem for these kernels (and their derivatives) under suitable domain and weight convergence. The results extend prior work on the reduced Bergman kernel and higher-order kernels to a broader weighted, higher-order setting, including eventually increasing domain sequences. These findings enhance understanding of regularity and asymptotic behavior of kernel functions in Dirichlet spaces and have potential applications in complex analysis on planar domains.

Abstract

For a planar domain $Ω$, we consider the Dirichlet spaces with respect to a base point $ζ\inΩ$ and the corresponding kernel functions. It is not known how these kernel functions behave as we vary the base point. In this note, we prove that these kernel functions vary smoothly. As an application of the smoothness result, we prove a Ramadanov-type theorem for these kernel functions on $Ω\timesΩ$. This extends the previously known convergence results of these kernel functions. In fact, we have made these observations in a more general setting, that is, for weighted kernel functions and their higher-order counterparts.

A Note on Kernel Functions of Dirichlet Spaces

TL;DR

The paper studies how the -th order weighted kernel functions , associated with Dirichlet spaces on planar domains, vary with the base point and the weight. It proves that is off a thin set , with explicit integral relations to the corresponding reduced Bergman kernels , and obtains a Ramadanov-type convergence theorem for these kernels (and their derivatives) under suitable domain and weight convergence. The results extend prior work on the reduced Bergman kernel and higher-order kernels to a broader weighted, higher-order setting, including eventually increasing domain sequences. These findings enhance understanding of regularity and asymptotic behavior of kernel functions in Dirichlet spaces and have potential applications in complex analysis on planar domains.

Abstract

For a planar domain , we consider the Dirichlet spaces with respect to a base point and the corresponding kernel functions. It is not known how these kernel functions behave as we vary the base point. In this note, we prove that these kernel functions vary smoothly. As an application of the smoothness result, we prove a Ramadanov-type theorem for these kernel functions on . This extends the previously known convergence results of these kernel functions. In fact, we have made these observations in a more general setting, that is, for weighted kernel functions and their higher-order counterparts.
Paper Structure (3 sections, 3 theorems, 36 equations)

This paper contains 3 sections, 3 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['smoothness']}
  3. Proof of Theorem \ref{['Ramadanov']}

Key Result

Theorem 1.5

Let $\Omega \subset \mathbb{C}$ be a domain, $\mu\in AW(\Omega)$ and $n$ be a positive integer. The kernel function $M_{\Omega,\mu}(\cdot, \cdot) \in C^{\infty}(\Omega\times\Omega)$. For $n>1$, the higher-order kernel function $M_{\Omega,\mu,n}(\cdot, \cdot) \in C^{\infty}(\Omega\times(\Omega\setmin where for a positive integer $m$

Theorems & Definitions (10)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Example 1
  • Theorem 1.6
  • Corollary 1.7
  • proof
  • proof