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H-harmonic reproducing kernels on the ball

Matěj Moravík

TL;DR

This work characterizes the Szegő reproducing kernel $K_h(x,y)$ for the $H$-harmonic Hardy space on the unit ball $B^n$ under the hyperbolic Laplacian $\Delta_h$. It derives an explicit series representation in terms of Exton's triple hypergeometric function $X_9$, with a compact corollary expressing $K_h$ as a finite sum of Gauss hypergeometric functions, and provides a specialized Appell function form when the arguments are linearly dependent via $y=\lambda x$. A parallel series for the weighted Bergman kernel $K_h^{s}$ is established, connecting to the standard $S_m$ and $Z_m$ expansions and reducing to $K_h$ when $I_m(s)=1$. The results yield closed-form and transform-based representations that enhance computability and facilitate analysis of $H$-harmonic kernels on the ball.

Abstract

We consider the Szegő reproducing kernel associated with the space of $H$-harmonic functions on the unit ball in n-dimensional space, i.e. functions that are characterized by being annihilated by the hyperbolic Laplacian. This paper derives an explicit series expansion for the reproducing kernel in terms of a triple hypergeometric function introduced of Exton. Moreover, we demonstrate that the Szegő kernel admits a representation as a finite sum of hypergeometric functions. We further show that the Szegő kernel, for linearly dependent arguments, can be expressed in terms of the first Appell hypergeometric function. In addition we provide a series expansion for the weighted Bergman kernels.

H-harmonic reproducing kernels on the ball

TL;DR

This work characterizes the Szegő reproducing kernel for the -harmonic Hardy space on the unit ball under the hyperbolic Laplacian . It derives an explicit series representation in terms of Exton's triple hypergeometric function , with a compact corollary expressing as a finite sum of Gauss hypergeometric functions, and provides a specialized Appell function form when the arguments are linearly dependent via . A parallel series for the weighted Bergman kernel is established, connecting to the standard and expansions and reducing to when . The results yield closed-form and transform-based representations that enhance computability and facilitate analysis of -harmonic kernels on the ball.

Abstract

We consider the Szegő reproducing kernel associated with the space of -harmonic functions on the unit ball in n-dimensional space, i.e. functions that are characterized by being annihilated by the hyperbolic Laplacian. This paper derives an explicit series expansion for the reproducing kernel in terms of a triple hypergeometric function introduced of Exton. Moreover, we demonstrate that the Szegő kernel admits a representation as a finite sum of hypergeometric functions. We further show that the Szegő kernel, for linearly dependent arguments, can be expressed in terms of the first Appell hypergeometric function. In addition we provide a series expansion for the weighted Bergman kernels.

Paper Structure

This paper contains 5 sections, 6 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Series expansion of the Szegö kernel
  4. Szegö kernel for linearly dependent arguments
  5. Bergman reproducing kernel

Key Result

Theorem 3.1

For each $\alpha > 1$, $\beta > 1$ and $n \geq 2$ we have where $X = \left(\frac{|x|}{1+|x|^2}\right)^2$, $Y =\frac{2\langle x,y\rangle}{(1+|y|^2)(1+|x|^2)}$ and $Z = \left(\frac{|y|}{1+|y|^2}\right)^2$.

Theorems & Definitions (14)

  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Proposition 4.1
  • Corollary 4.1
  • proof
  • ...and 4 more