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Supercontractivity and ultracontractivity for (nonsymmetric) diffusion semigroups on manifolds

Supercontractivity and ultracontractivity for (nonsymmetric) diffusion semigroups on manifolds Abstract. Under a curvature condition, we obtain some criteria of supercontractivity and ultracontractivity for (non-symmetric) di¤usion semigroups on Riemannian manifolds. As consequences, some sucient and necessary conditions are presented for these two contraction properties. Then, based on a perturbation argument, ultracontractivity is studied for symmetric di¤usions with non-smooth drifts. Moreover, we provide a way to study contraction properties using isoperimetric inequalities. To illustrate the main results, some typical examples are constructed, where one of them shows that supercontractivity is strictly weaker than ultracontractivity as already known in the flat case. 1991 Mathematics Subject Classification: 35J10, 60J60. 1 Introduction Let M be a d-dimensional, connected, noncompact, complete Riemannian manifold with boundary qM either convex or empty. Denote by dx the Riemannian volume element. Let m be a positive Radon measure on M and Pt a (sub) Markov semigroup on L 2 ðmÞ. We call Pt ultracontractive (respectively, supercontractive) if kPt k2!y < y (respectively, kPt k2!4 < y) for any t > 0, where k Á kp!q denotes the operator norm from L p ðmÞ to L q ðmÞ. When Pt is symmetric on L 2 ðmÞ, kPt k1!y < y for all t > 0 is equivalent to http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Supercontractivity and ultracontractivity for (nonsymmetric) diffusion semigroups on manifolds

Forum Mathematicum , Volume 15 (6) – Oct 1, 2003

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References (46)

Publisher
de Gruyter
Copyright
Copyright © 2003 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2003.044
Publisher site
See Article on Publisher Site

Abstract

Abstract. Under a curvature condition, we obtain some criteria of supercontractivity and ultracontractivity for (non-symmetric) di¤usion semigroups on Riemannian manifolds. As consequences, some sucient and necessary conditions are presented for these two contraction properties. Then, based on a perturbation argument, ultracontractivity is studied for symmetric di¤usions with non-smooth drifts. Moreover, we provide a way to study contraction properties using isoperimetric inequalities. To illustrate the main results, some typical examples are constructed, where one of them shows that supercontractivity is strictly weaker than ultracontractivity as already known in the flat case. 1991 Mathematics Subject Classification: 35J10, 60J60. 1 Introduction Let M be a d-dimensional, connected, noncompact, complete Riemannian manifold with boundary qM either convex or empty. Denote by dx the Riemannian volume element. Let m be a positive Radon measure on M and Pt a (sub) Markov semigroup on L 2 ðmÞ. We call Pt ultracontractive (respectively, supercontractive) if kPt k2!y < y (respectively, kPt k2!4 < y) for any t > 0, where k Á kp!q denotes the operator norm from L p ðmÞ to L q ðmÞ. When Pt is symmetric on L 2 ðmÞ, kPt k1!y < y for all t > 0 is equivalent to

Journal

Forum Mathematicumde Gruyter

Published: Oct 1, 2003

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