Hardy type spaces on certain noncompact manifolds and applications

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.

27 pages, v2: the first version has been revised and rearranged, with additions, in two papers, of which this new version is the first. The second paper is posted as arXiv:1002.1161v1

References in corpus (4)

• Hardy type spaces on certain noncompact manifolds and applications
• H^1 and BMO for certain locally doubling metric measure spaces of finite measure
• H^1 and BMO for certain nondoubling metric measure spaces
• Estimates for functions of the Laplacian on manifolds with bounded geometry

Cited by in corpus (8)

• Hardy type spaces on certain noncompact manifolds and applications
• Higher order Riesz transforms on noncompact symmetric spaces
• A family of Hardy type spaces on nondoubling manifolds
• Local Hardy Spaces of Differential Forms on Riemannian Manifolds
• Inclusions and noninclusions of Hardy type spaces on certain nondoubling manifolds
• Harmonic Bergman spaces, the Poisson equation and the dual of Hardy-type spaces on certain noncompact manifolds
• Atomic decomposition of Hardy type spaces on certain noncompact manifolds
• Sharp endpoint results for imaginary powers and Riesz transforms on certain noncompact manifolds