Riesz transform on manifolds and Poincaré inequalitie

Pascal Auscher; Thierry Coulhon

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Riesz transform on manifolds and heat kernel regularity

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Assume that $M_{0}$ is a complete Riemannian manifold with Ricci curvature bounded from below and that $M_{0}$ satisfies a Sobolev inequality of dimension $ν > 3$ . Let $M$ be a complete Riemannian manifold isometric at infinity to $M_{0}$ and let $p \in (ν / (ν - 1), ν)$ . The boundedness of the Riesz transform of $L^{p} (M_{0})$ then implies the boundedness of the Riesz transform of $L^{p} (M)$

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Weak-type estimates for the Riesz transforms associated with the Gaussian measure.

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In this paper we will study the behavior of the Riesz transform associated with the Gaussian measure γ(x)dx = edx in the space L (R).

Metric properties of eigenfunctions of the Laplace operator on manifolds

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On a two-dimensional compact real analytic Riemannian manifold we estimate the volume of the set on which the eigenfunction of the Laplace-Beltrami operator is positive. On an $n$ -dimensional compact smooth Riemannian manifold, we estimate the relation between supremum and infimum of an eigenfunction of the Laplace operator.