Harmonic functions for Bessel operators

TL;DR

This paper investigates the boundedness of the Riesz transform associated with Bessel operators on certain manifolds, extending classical harmonic analysis results to more general geometric settings.

Contribution

It extends the classical theory of Riesz transforms to Bessel operators on manifolds with ends, providing new insights into harmonic analysis in non-Euclidean contexts.

Findings

Proves the continuity of the Riesz transform from Hardy space to L^1 for Bessel operators.

Extends classical Euclidean results to manifolds with ends and Bessel-type measures.

Provides a framework for analyzing harmonic analysis operators on non-standard geometries.

Abstract

We verify the continuity of the Riesz transform from the operator related Hardy space to $L^1$ - Lebesgue space of integrable functions. For the standard Euclidean Laplace operator, this is a classical result that plays a significant role in harmonic analysis and theory of singular integral operators. Here, we consider a one-dimensional model of manifolds with ends and external Dirichlet boundary operators. This setting extends the work of Hassell and the third author. Specifically, we examine the real line with the measure $|x|^{n-1}dx$ leading to various versions of Bessel operators. For integer $n$, this mimics the measure on Euclidean $n$-dimensional space and the obtained results are expected to provide good predictions for a class of Riemannian manifolds with Euclidean ends.