Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem

Coulhon, Thierry; Sikora, Adam

doi:10.1112/plms/pdm050

Cited by 176 publications

(170 citation statements)

References 52 publications

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“…In this paper, we assume that µ(X) = ∞. It is not difficult to see that the condition (8) implies that there exists a constant n ≥ 0 so that…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

“…Moreover, there exists a positive constant C such that |B((x, t), r)| = Cr Q , where Q = 2d + 2 is the homogeneous dimension of H n and |B((x, t), r)| is the Lebesgue measure of the ball B((x, t), r). Obviously, the triplet (H n , d, dx) satisfies the doubling condition (8).…”

Section: 4mentioning

confidence: 99%

“…In the first direction Coifman and Weiss [9] introduced H p at (X) on a space X of homogeneous type (see (8) below) and gave versions of (1) under further geometric conditions on X. Whether something like (1) holds without any extra condition on X is still open, but the case when X has 'reverse doubling' has been solved in [40,43] for p ∈ (p 0 , 1] with certain p 0 ∈ (0, 1).…”

Section: Introductionmentioning

confidence: 99%

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Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type

Duong

2018

Trans. Amer. Math. Soc.

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Abstract. Let X be a space of homogeneous type and let L be a nonnegative self-adjoint operator on L 2 (X) enjoying Gaussian estimates. The main aim of this paper is twofold. Firstly, we prove (local) nontangential and radial maximal function characterizations for the local Hardy spaces associated to L. This gives the maximal function characterization for local Hardy spaces in the sense of Coifman and Weiss provided that L satisfies certain extra conditions. Secondly we introduce local Hardy spaces associated with a critical function ρ which are motivated by the theory of Hardy spaces related to Schrödinger operators and of which include the local Hardy spaces of Coifman and Weiss as a special case. We then prove that these local Hardy spaces can be characterized by (local) nontangential and radial maximal functions related to L and ρ, and by global maximal functions associated to 'perturbations' of L. We apply our theory to obtain a number of new results on maximal characterizations for the local Hardy type spaces in various settings ranging from Schrödinger operators on manifolds to Schrödinger operators on connected and simply connected nilpotent Lie groups.

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“…In this paper, we assume that µ(X) = ∞. It is not difficult to see that the condition (8) implies that there exists a constant n ≥ 0 so that…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type

Duong

2018

Trans. Amer. Math. Soc.

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“…Further we shall use the following lemma, whose proof based on finite speed propagation of the wave equation (see [9]) can be found in [11]. …”

Section: Corollary 7 There Is a Constant Cmentioning

confidence: 99%

Hardy spaces for semigroups with Gaussian bounds

Dziubański

Preisner

2017

Annali di Matematica

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“…In particular, the operator cos(t √ L) is then welldefined and bounded on L 2 (R n ). Moreover, it follows from Theorem 3 of [13] that if the corresponding heat kernels p t (x, y) of e −tL satisfy Gaussian bounds (GE), then there exists a finite, positive constant c 0 with the property that the Schwartz kernel…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Weighted L p estimates for the area integral associated to self-adjoint operators

Gong

Yan

2013

manuscripta math.

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Abstract. This article is concerned with some weighted norm inequalities for the so-called horizontal (i.e. involving time derivatives) area integrals associated to a non-negative selfadjoint operator satisfying a pointwise Gaussian estimate for its heat kernel, as well as the corresponding vertical (i.e. involving space derivatives) area integrals associated to a non-negative self-adjoint operator satisfying in addition a pointwise upper bounds for the gradient of the heat kernel. As applications, we obtain sharp estimates for the operator norm of the area integrals on L p

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Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem

Abstract: Abstract. We prove that in presence of L 2 Gaussian estimates, so-called Davies-Gaffney estimates, on-diagonal upper bounds imply precise off-diagonal Gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces.

Cited by 176 publications

References 52 publications

Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type

Maximal function characterizations for new local Hardy-type spaces on spaces of homogeneous type

Hardy spaces for semigroups with Gaussian bounds

Weighted L p estimates for the area integral associated to self-adjoint operators

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