The weighted Bergman space on a sector and a degenerate parabolic equation

TL;DR

This paper solves a degenerate parabolic equation on the half line, demonstrating that its null reachable space forms a Reproducing Kernel Hilbert Space of analytic functions on a sector, characterized by a specific reproducing kernel.

Contribution

It introduces a novel connection between degenerate parabolic equations and weighted Bergman spaces via RKHS theory, providing explicit kernel representations.

Findings

Null reachable space is a RKHS of analytic functions on a sector

Reproducing kernel expressed via weighted Bergman kernel on C+

Solution methodology for degenerate parabolic equations

Abstract

In this work we solve a degenerate parabolic equation for the half line with Dirichlet boundary data, and use some results from the theory of Reproducing Kernel Hilbert Spaces to show that the null reachable space of this degenerate parabolic equation is a RKHS of analytic functions on a sector, whose reproducing kernel can be written in terms of the weighted Bergman kernel on the half plane C+.