413 research outputs found
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
Partial symmetry and existence of least energy solutions to some nonlinear elliptic equations on Riemannian models
We consider least energy solutions to the nonlinear equation posed on a class of Riemannian models of dimension
which include the classical hyperbolic space as well as manifolds
with unbounded sectional geometry. Partial symmetry and existence of least
energy solutions is proved for quite general nonlinearities , where
denotes the geodesic distance from the pole of
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
A family of Hardy-type spaces on nondoubling manifolds
We introduce a decreasing one-parameter family Xγ(M) , γ> 0 , of Banach subspaces of the Hardy–Goldberg space h1(M) on certain nondoubling Riemannian manifolds with bounded geometry, and we investigate their properties. In particular, we prove that X1 / 2(M) agrees with the space of all functions in h1(M) whose Riesz transform is in L1(M) , and we obtain the surprising result that this space does not admit an atomic decomposition
Maximal characterisation of local Hardy spaces on locally doubling manifolds
We prove a radial maximal function characterisation of the local atomic Hardy space h1(M) on a Riemannian manifold M with positive injectivity radius and Ricci curvature bounded from below. As a consequence, we show that an integrable function belongs to h1(M) if and only if either its local heat maximal function or its local Poisson maximal function is integrable. A key ingredient is a decomposition of Hölder cut-offs in terms of an appropriate class of approximations of the identity, which we obtain on arbitrary Ahlfors-regular metric measure spaces and generalises a previous result of A. Uchiyama
Analysis on Trees with Nondoubling Flow Measures
We consider trees with root at infinity endowed with flow measures, which are nondoubling measures of at least exponential growth and which do not satisfy the isoperimetric inequality. In this setting, we develop a Calderón–Zygmund theory and we define BMO and Hardy spaces, proving a number of desired results extending the corresponding theory as known in more classical settings
BMO Spaces on Weighted Homogeneous Trees
We consider an infinite homogeneous tree V endowed with the usual metric d defined on graphs and a weighted measure μ. The metric measure space (V, d, μ) is nondoubling and of exponential growth, hence the classical theory of Hardy and BMO spaces does not apply in this setting. We introduce a space BMO(μ) on (V, d, μ) and investigate some of its properties. We prove in particular that BMO(μ) can be identified with the dual of a Hardy space H1(μ) introduced in a previous work and we investigate the sharp maximal function related with BMO(μ)
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