Hardy type spaces on certain noncompact manifolds and applications

Details Hardy type spaces on certain noncompact manifolds and applications Hardy type spaces on certain noncompact manifolds and applications

2008-12-22

arxiv.org/abs/0812.4209v2

Mathematics

Functional Analysis

27 pages, v2: the first version has been revised and rearranged, with additions, in two papers, of which this new version is t

Scientific paper

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.

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