Inclusions and noninclusions of Hardy type spaces on certain nondoubling manifolds

TL;DR

This paper investigates the relationships between various Hardy type spaces on certain noncompact, nondoubling manifolds, revealing new isomorphisms and distinctions that contrast with classical Euclidean results.

Contribution

It establishes isomorphic relations between Riesz-Hardy and Goldberg type spaces on manifolds, and demonstrates the nonexistence of atomic characterizations, also highlighting differences among Hardy spaces on Damek-Ricci spaces.

Findings

$H^1_ R(M)$ is isomorphic to $rak{h}^1(M)$ via a specific operator

$rak{h}^1(M)$ lacks an atomic characterization

On Damek-Ricci spaces, different Hardy spaces are mutually distinct

Abstract

In this paper we establish inclusions and noninclusions between various Hardy type spaces on noncompact Riemannian manifolds $M$ with Ricci curvature bounded from below, positive injectivity radius and spectral gap. Our first main result states that, if $\mathscr{L}$ is the positive Laplace-Beltrami operator on $M$, then the Riesz-Hardy space $H^1_\mathscr{R}(M)$ is the isomorphic image of the Goldberg type space $\mathfrak{h}^1(M)$ via the map $\mathscr{L}^{1/2} (\mathscr{I} + \mathscr{L})^{-1/2}$, a fact that is false in $\mathbb{R}^n$. Specifically, $H^1_\mathscr{R}(M)$ agrees with the Hardy type space $\mathfrak{X}^{1/2}(M)$ recently introduced by the the first three authors; as a consequence, we prove that $\mathfrak{h}^1(M)$ does not admit an atomic characterisation. Noninclusions are mostly proved in the special case where the manifold is a Damek-Ricci space $S$. Our second main result states that $H^1_\mathscr{R}(S)$, the heat Hardy space $H^1_\mathscr{H}(S)$ and the Poisson-Hardy space $H^1_\mathscr{P}(S)$ are mutually distinct spaces, a fact which is in sharp contrast to the Euclidean case, where these three spaces agree.