385 research outputs found
Hardy type spaces on certain noncompact manifolds and applications
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.Comment: 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:1002.1161v
Higher order Riesz transforms on noncompact symmetric spaces
In this note we prove various sharp boundedness results on suitable Hardy
type spaces for Riesz transforms of arbitrary order on noncompact symmetric
spaces of arbitrary rank.Comment: v2: the first version has been revised and splitted up in two papers,
of which this new version is one par
Estimates for functions of the Laplacian on manifolds with bounded geometry
In this paper we consider a complete connected noncompact Riemannian manifold
M with Ricci curvature bounded from below and positive injectivity radius.
Denote by L the Laplace-Beltrami operator on M. We assume that the kernel
associated to the heat semigroup generated by L satisfies a mild decay
condition at infinity. We prove that if m is a bounded holomorphic function in
a suitable strip of the complex plane, and satisfies Mihlin-Hormander type
conditions of appropriate order at infinity, then the operator m(L) extends to
an operator of weak type 1.
This partially extends a celebrated result of J. Cheeger, M. Gromov and M.
Taylor, who proved similar results under much stronger curvature assumptions on
M, but without any assumption on the decay of the heat kernel.Comment: 19 page
Dispersive estimates with loss of derivatives via the heat semigroup and the wave operator
This paper aims to give a general (possibly compact or noncompact) analog of
Strichartz inequalities with loss of derivatives, obtained by Burq, G\'erard,
and Tzvetkov [19] and Staffilani and Tataru [51]. Moreover we present a new
approach, relying only on the heat semigroup in order to understand the
analytic connexion between the heat semigroup and the unitary Schr\"odinger
group (both related to a same self-adjoint operator). One of the novelty is to
forget the endpoint dispersive estimates and to look for a
weaker estimates (Hardy and BMO spaces both adapted to the heat
semigroup). This new point of view allows us to give a general framework
(infinite metric spaces, Riemannian manifolds with rough metric, manifolds with
boundary,...) where Strichartz inequalities with loss of derivatives can be
reduced to microlocalized dispersive properties. We also use the link
between the wave propagator and the unitary Schr\"odinger group to prove how
short time dispersion for waves implies dispersion for the Schr\"odinger group.Comment: 48 page
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