A family of Hardy type spaces on nondoubling manifolds

TL;DR

This paper introduces a new family of Hardy-type spaces on nondoubling manifolds, exploring their properties and revealing that one such space coincides with functions having Riesz transforms in L^1, but lacks atomic decomposition.

Contribution

It defines a novel family of Hardy spaces on nondoubling manifolds and characterizes a specific space within this family, showing it does not admit atomic decomposition.

Findings

The space $rak{X}^{1/2}(M)$ equals functions in $rak{h}^1(M)$ with Riesz transform in L^1.

The space $rak{X}^{1/2}(M)$ does not admit an atomic decomposition.

The family $rak{X}^{eta}(M)$ provides a new framework for Hardy spaces on nondoubling manifolds.

Abstract

We introduce a decreasing one-parameter family $\mathfrak{X}^{\gamma}(M)$, $\gamma>0$, of Banach subspaces of the Hardy-Goldberg space $\mathfrak{h}^1(M)$ on certain nondoubling Riemannian manifolds with bounded geometry and we investigate their properties. In particular, we prove that $\mathfrak{X}^{1/2}(M)$ agrees with the space of all functions in $\mathfrak{h}^1(M)$ whose Riesz transform is in $L^1(M)$, and we obtain the surprising result that this space does not admit an atomic decomposition.