Sharp endpoint results for imaginary powers and Riesz transforms on certain noncompact manifolds

TL;DR

This paper establishes boundedness results for imaginary powers of the Laplacian and Riesz transforms on Hardy spaces over certain noncompact manifolds, advancing harmonic analysis in geometric contexts.

Contribution

It proves boundedness of these operators from Hardy space X^1(M) to L^1(M) on manifolds with bounded geometry and spectral gap, extending previous work.

Findings

Imaginary powers of the Laplacian are bounded from X^1(M) to L^1(M)

Riesz transforms are bounded from X^1(M) to L^1(M)

Results apply to noncompact manifolds with bounded geometry and spectral gap

Abstract

In this paper we consider a complete connected noncompact Riemannian manifold M with bounded geometry and spectral gap. We prove that the imaginary powers of the Laplacian and the Riesz transform are bounded from the Hardy space X^1(M), introduced in previous work of the authors, to L^1(M).