Uniformly Elliptic Operators with Measurable Coefficients

Davies, E. B.

doi:10.1006/jfan.1995.1103

Cited by 164 publications

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“…The inequality for the first term of the left-hand side is classical (see, e.g., [27]). The estimate for the second one follows from essentially the same proof (see [28], Lemma 7). We give a proof for the third one as we could not find it in the literature in this situation (it is proved in [4] for elliptic operators on R n ).…”

Section: Proof Of the Main Resultsmentioning

confidence: 77%

Riesz transform on manifolds and heat kernel regularity

Auscher

Coulhon

Duong

et al. 2004

Annales Scientifiques de l’École Normale Supérieure

219

413

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On considère la classe des variétés riemanniennes complètes non compactes dont le noyau de la chaleur satisfait une estimation supérieure et inférieure gaussienne. On montre que la transformée de Riesz y est bornée sur L p , pour un intervalle ouvert de p au-dessus de 2, si et seulement si le gradient du noyau de la chaleur satisfait une certaine estimation L p pour le même intervalle d'exposants p.One considers the class of complete non-compact Riemannian manifolds whose heat kernel satisfies Gaussian estimates from above and below. One shows that the Riesz transform is L p bounded on such a manifold, for p ranging in an open interval above 2, if and only if the gradient of the heat kernel satisfies a certain L p estimate in the same interval of p's.MSC numbers 2000: 58J35, 42B20

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Section: Proof Of the Main Resultsmentioning

confidence: 77%

Riesz transform on manifolds and heat kernel regularity

Auscher

Coulhon

Duong

et al. 2004

Annales Scientifiques de l’École Normale Supérieure

219

413

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“…In particular, Lemma 4.8 is a consequence of Lemma 4.9 and is a generalization of results from [5,15] (a) q = 2, P(p), P(q) and K(p) imply P(r), and p = 2, P*(/? ), P*(q) and K*(q) imply P*(r); …”

Section: Off-diagonal Estimatesmentioning

confidence: 78%

Extrapolation of the functional calculus of generalized Dirac operators and related embedding and Littlewood-Paley-type theorems. I

Ajiev¹

2007

J. Aust. Math. Soc.

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Several rather general sufficient conditions for the extrapolation of the calculus of generalized Dirac operators from t j to L p are established. As consequences, we obtain some embedding theorems, quadratic estimates and Littlewood-Paley theorems in terms of this calculus in Lebesgue spaces. Some further generalizations, utilised in Part II devoted to applications, which include the Kato square root model, are discussed. We use resolvent approach and show the irrelevance of the semigroup one. Auxiliary results include a high order counterpart of the Hilbert identity, the derivation of new forms of 'off-diagonal' estimates, and the study of the structure of the model in Lebesgue spaces and its interpolation properties. In particular, some coercivity conditions for forms in Banach spaces are used as a substitution of the ellipticity ones. Attention is devoted to the relations between the properties of perturbed and unperturbed generalized Dirac operators. We do not use any stability results.2000 Mathematics subject classification: primary 46E15, 46E30. 47A60. 47A65: secondary 47A05, 47A55, 46B20. 46C99.42C99.

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“…Davies perturbation estimates are well known for various classes of elliptic or subelliptic operators on manifolds: see, for example, [5], [14] and [19]. In our situation, however, the drift term A 0 creates new and non-trivial difficulties in the proof.…”

Section: Introductionmentioning

confidence: 84%

“…which implies the desired norm estimate (compare, for example, [5,Section 4]). Thus we concentrate on proving the first estimate of the theorem.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

Heat kernel and semigroup estimates for sublaplacians with drift on Lie groups

Dungey¹

2005

Publicacions Matemàtiques

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Let G be a Lie group. The main new result of this paper is an estimate in L 2 (G) for the Davies perturbation of the semigroup generated by a centered sublaplacian H on G. When G is amenable, such estimates hold only for sublaplacians which are centered. Our semigroup estimate enables us to give new proofs of Gaussian heat kernel estimates established by Varopoulos on amenable Lie groups and by Alexopoulos on Lie groups of polynomial growth.

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Uniformly Elliptic Operators with Measurable Coefficients

Cited by 164 publications

References 0 publications

Riesz transform on manifolds and heat kernel regularity

Riesz transform on manifolds and heat kernel regularity

Extrapolation of the functional calculus of generalized Dirac operators and related embedding and Littlewood-Paley-type theorems. I

Heat kernel and semigroup estimates for sublaplacians with drift on Lie groups

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