Atomic decomposition of Hardy type spaces on certain noncompact manifolds

TL;DR

This paper establishes an atomic decomposition for Hardy type spaces on certain noncompact manifolds and demonstrates boundedness of higher-order Riesz transforms on these spaces, advancing harmonic analysis in geometric contexts.

Contribution

It provides the first atomic characterization of Hardy type spaces on noncompact manifolds with bounded geometry and spectral gap, and proves boundedness of higher-order Riesz transforms.

Findings

Atomic decomposition of X^k(M) spaces established

Higher-order Riesz transforms are bounded on X^k(M) and L^p(M)

Results extend harmonic analysis tools to noncompact manifolds

Abstract

In this paper we consider a complete connected noncompact Riemannian manifold M with bounded geometry and spectral gap. We prove that the Hardy type spaces X^k(M), introduced in a previous paper of the authors, have an atomic characterization. As an application, we prove that the Riesz transforms of even order 2k are bounded from X^k(M) to L^1(M)and on L^p(M) for 1<p<\infty.