arXiv subjects

Aakanksha Jain

Publications and source records attributed to Aakanksha Jain.

7 recordsLinked to original sources

arXiv2026

A Canonical Positive Definite Kernel Associated with the $\xi$-Bergman Kernel

Let $\Omega \subset \mathbb{C}^{n}$ and $\xi \in \ell^{1}$. The $\xi$-Bergman kernel $K_{\xi, \Omega}$, introduced by Bao and Guan, generalizes the classical Bergman kernel by replacing the point evaluation functional with a functional determined by sequence $\xi$. While this kernel inherits several important extremal and plurisubharmonic properties, it is intrinsically an on-diagonal object and therefore lacks the two-variable reproducing kernel structure that lies at the heart of the classical Bergman theory. The purpose of this paper is to associate a canonical Hermitian positive-definite kernel with the $\xi$-Bergman kernel and to investigate its analytic and geometric properties. Our construction is based on the family of Riesz representatives corresponding to the $\xi$-evaluation functionals. More precisely, we introduce a Hermitian kernel obtained as the Gram kernel of these representatives and show that it is positive definite and for $z \in \Omega$ satisfies \[ B_{\xi,\Omega}(z,z)=K_{\xi,\Omega}(z), \] thereby recovering the $\xi$-Bergman kernel as its diagonal restriction. As a consequence, we prove that the $\xi$-Bergman kernel is real analytic on $\Omega$. We also establish biholomorphic transformation laws, and obtain a representation of the $\xi$-Bergman kernel in terms of derivatives of the classical Bergman kernel. Furthermore, we obtain explicit formulas for the $\xi$-Bergman kernel on the upper half-plane $\mathbb{H}$ corresponding to several classes of sequences $\xi$, establish corresponding $\xi$-Lu Qi-Keng results, and derive precise boundary asymptotics. These examples illustrate how the choice of the differential functional influences both the zero set and the boundary growth of the associated kernel.

math.CV↗

arXiv2025

Weighted Kernel Functions on Planar Domains

We study the variation of weighted Szegő and Garabedian kernels on planar domains as a function of the weight. A Ramadanov type theorem is shown to hold as the weights vary. As a consequence, we derive properties of the zeros of the weighted Szegő and Garabedian kernel for weights close to the constant function $1$ on the boundary. We further study the weighted Ahlfors map and strengthen results concerning its boundary behaviour. Explicit examples of the weighted kernels are presented for certain classes of weights. We highlight an interesting property of the weighted Szegő and Garabedian kernels, implicit in Nehari's work, and explore several of its consequences. Finally, we discuss the weighted Carathéodory metric, and describe relations of the weighted Szegő and Garabedian kernel with certain classical kernel functions.

math.CV↗

arXiv2025

Weighted Szegő Kernels on Planar Domains

We study 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), it is shown that some properties of the unweighted Szegő kernel propagate to the weighted Szegő kernel as well. Finally, it is shown 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.

math.CV↗

arXiv2024

Weighted Bergman Kernels on Planar Domains

Boundary Behaviour of Weighted Bergman Kernels: For a planar domain $D \subset \mathbb{C}$ and an admissible weight function $μ$ on it, some aspects of the boundary behaviour of the corresponding weighted Bergman kernel $K_{D, μ}$ are studied. First, under the assumption that $μ$ extends continuously to a smooth boundary point $p$ of $D$ and is non-vanishing there, we obtain a precise relation between $K_{D, μ}$ and the classical Bergman kernel $K_D$ near $p$. Second, when viewed as functions of such weights, the weighted Bergman kernel is shown to have a suitable additive and multiplicative property near such boundary points. A Study on Holomorphic Isometries of Weighted Bergman Metrics: For a domain $D \subset \mathbb{C}^n$ and an admissible weight $μ$ on it, we consider the weighted Bergman kernel $K_{D, μ}$ and the corresponding weighted Bergman metric on $D$. In particular, motivated by work of Mok, Ng, Chan--Yuan and Chan--Xiao--Yuan among others, we study 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 $μ= K_{\mathbb{D}}^{-d}$ for some integer $d \geq 0$. These metrics provide a natural class of examples that give rise to positive conformal constants that have been considered in various recent works on isometries. Specific examples of isometries that are studied 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, the case of isometries between polydisks in possibly different dimensions, in which each factor has a different weighted Bergman metric as above, is also presented.

math.CV↗

arXiv2024

A Note on Kernel Functions of Dirichlet Spaces

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.

math.CV↗

arXiv2023

The Reduced Bergman Kernel and its Properties

In this article, we study some properties of the $n$-th order weighted reduced Bergman kernels for planar domains, $n\geq 1$. Specifically, we look at Ramadanov type theorems, localization, and boundary behaviour of the weighted reduced Bergman kernel and its higher-order counterparts. We also give a transformation formula for these kernels under biholomorphisms.

math.CV↗

arXiv2022

Transformation formula for the Reduced Bergman kernel and its Application

In this article, we prove the transformation formula for the reduced Bergman kernels under proper holomorphic correspondences between bounded domains in the complex plane. As a corollary, we obtain the transformation formula for the reduced Bergman kernels under proper holomorphic maps. We also establish the transformation formula for the weighted reduced Bergman kernels under proper holomorphic maps. Finally, we provide an application of this transformation formula.

math.CV↗