Ultracontractivity and embedding into L ∞

A. Bendikov; T. Coulhon; L. Saloff-Coste

doi:10.1007/s00208-006-0057-z

Ultracontractivity and embedding into L ^∞

A. Bendikov
, T. Coulhon
, L. Saloff-Coste

Wroclaw University
CY Cergy Paris Université
Cornell University

Research output: Contribution to journal › Article › peer-review

Abstract

Given a self-adjoint semigroup e^-tA satisfying an ultracontractivity bound of the type ∥e^-tA∥2→∞ ≤ e^m(t), we find conditions on the sequence ∥A ⁿf∥₂^1/n that imply that f is a bounded function. Sobolev's classical embedding theorem says that, when A is the Laplace operator on ℝ^d, ∥A^kf∥2 < ∞ for some k>d/4 suffices to imply that f is bounded. In the cases we are interested in, the desired condition involves the whole sequence ∥Aⁿf∥ ₂^1/n and depends on the behavior of the ultracontractivity function m.

Original language	English
Pages (from-to)	817-853
Number of pages	37
Journal	Mathematische Annalen
Volume	337
Issue number	4
DOIs	https://doi.org/10.1007/s00208-006-0057-z
Publication status	Published - 1 Apr 2007
Externally published	Yes

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title = "Ultracontractivity and embedding into L ∞",

abstract = "Given a self-adjoint semigroup e-tA satisfying an ultracontractivity bound of the type ∥e-tA∥2→∞ ≤ em(t), we find conditions on the sequence ∥A nf∥21/n that imply that f is a bounded function. Sobolev's classical embedding theorem says that, when A is the Laplace operator on ℝd, ∥Akf∥2 < ∞ for some k>d/4 suffices to imply that f is bounded. In the cases we are interested in, the desired condition involves the whole sequence ∥Anf∥ 21/n and depends on the behavior of the ultracontractivity function m.",

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AU - Coulhon, T.

AU - Saloff-Coste, L.

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N2 - Given a self-adjoint semigroup e-tA satisfying an ultracontractivity bound of the type ∥e-tA∥2→∞ ≤ em(t), we find conditions on the sequence ∥A nf∥21/n that imply that f is a bounded function. Sobolev's classical embedding theorem says that, when A is the Laplace operator on ℝd, ∥Akf∥2 < ∞ for some k>d/4 suffices to imply that f is bounded. In the cases we are interested in, the desired condition involves the whole sequence ∥Anf∥ 21/n and depends on the behavior of the ultracontractivity function m.

AB - Given a self-adjoint semigroup e-tA satisfying an ultracontractivity bound of the type ∥e-tA∥2→∞ ≤ em(t), we find conditions on the sequence ∥A nf∥21/n that imply that f is a bounded function. Sobolev's classical embedding theorem says that, when A is the Laplace operator on ℝd, ∥Akf∥2 < ∞ for some k>d/4 suffices to imply that f is bounded. In the cases we are interested in, the desired condition involves the whole sequence ∥Anf∥ 21/n and depends on the behavior of the ultracontractivity function m.

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JO - Mathematische Annalen

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ER -

Ultracontractivity and embedding into L ∞

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Ultracontractivity and embedding into L ^∞