Local Riesz transform and local Hardy spaces on Riemannian manifolds with bounded geometry

TL;DR

This paper establishes the equivalence of certain local Hardy spaces and Riesz transform spaces on Riemannian manifolds with bounded geometry, and relates these to harmonic functions on a product space.

Contribution

It proves the equivalence of Goldberg-type and Riesz transform-based local Hardy spaces on manifolds with bounded geometry, extending harmonic analysis tools to this setting.

Findings

The spaces nd nd re shown to be identical under certain conditions.

A relationship between nd harmonic functions on a product space is established.

The results apply to complete noncompact Riemannian manifolds with Ricci curvature bounded below.

Abstract

We prove that if $\tau$ is a large positive number, then the atomic Goldberg-type space $\mathfrak{h}^1(N)$ and the space $\mathfrak{h}_{\mathcal R_\tau}^1(N)$ of all integrable functions on $N$ whose local Riesz transform $\mathcal R_\tau$ is integrable are the same space on any complete noncompact Riemannian manifold $N$ with Ricci curvature bounded from below and positive injectivity radius. We also relate $\mathfrak{h}^1(N)$ to a space of harmonic functions on the slice $N\times (0,\delta)$ for $\delta>0$ small enough.