Hardy type spaces on certain noncompact manifolds and applications

TL;DR

This paper introduces new Hardy spaces on certain noncompact manifolds with Ricci curvature bounds, enabling endpoint estimates for spectral multipliers and Riesz transforms, with applications to symmetric spaces.

Contribution

It develops a novel Hardy space framework on noncompact manifolds, facilitating endpoint spectral multiplier estimates and Riesz transform bounds.

Findings

Established Hardy spaces X^k(M) and duals Y^k(M) for noncompact manifolds.

Derived endpoint estimates for imaginary powers of Laplace-Beltrami operator.

Proved endpoint results for Riesz transforms under volume growth conditions.

Abstract

In this paper we consider a complete connected noncompact Riemannian manifold M with Ricci curvature bounded from below, positive injectivity radius and spectral gap b. We introduce a sequence X^1(M), X^2(M), ... of new Hardy spaces on M, the sequence Y^1(M/, Y^2(M), ... of their dual spaces, and show that these spaces may be used to obtain endpoint estimates for purely imaginary powers of the Laplace-Beltrami operator and for more general spectral multipliers associated to the Laplace--Beltrami operator L on M. Under the additional condition that the volume of the geodesic balls of radius r is controlled by C r^a e^{2\sqrt{b} r} for some real number a and for all large r, we prove also an endpoint result for first order Riesz transforms D L^{-1/2}. In particular, these results apply to Riemannian symmetric spaces of the noncompact type.