Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians

Davies, E. B.; Simon, Barry

doi:10.1016/0022-1236(84)90076-4

Cited by 370 publications

(314 citation statements)

References 24 publications

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“…This has been shown in Davies and Simon (1984), Aida et al(1994) and Ledoux (1995) and we recall the simple steps here. In the recent note Bobkov (1995), Talagrand's inequality (1.2) on the cube is deduced in this way from Gross's logarithmic Sobolev inequality on the two point space.…”

Section: T Umentioning

confidence: 85%

On Talagrand's deviation inequalities for product measures

1997

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Abstract. We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M . Talagrand in the recent y ears. Our method is based on functional inequalities of Poincar e and logarithmic Sobolev type and iteration of these inequalities. In particular, we establish with these tools sharp deviation inequalities from the mean on norms of sums of independent random vectors and empirical processes. Concentration for the Hamming distance may also be deduced from this approach.

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Section: T Umentioning

confidence: 85%

On Talagrand's deviation inequalities for product measures

1997

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“…The existence of l λ α (t, x, y) can be proved arguing as in [DS1]; that is, using a global Sobolev inequality on Ω, which can be easily deduced from its local version (2.12) as well as (2.18), by means of the partition of unity as in [K].…”

Section: Above Inequality Can Be Restated As Follows I(i + 1) ≤ C(i)imentioning

confidence: 99%

“…Due to the results in Lemma 7 in [DD] and using Theorem 7.1 in [DS1] on one hand and elliptic regularity on the other, there exist two positive constants c 1 , c 2 such that…”

Section: Proof Of Theorem 34mentioning

confidence: 99%

Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators on Bounded Domains

Filippas

Moschini

Tertikas

2007

Commun. Math. Phys.

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On a smooth bounded domain Ω ⊂ R N we consider the Schrödinger operators −∆−V , with V being either the critical borderline potential V (x) = (N − 2) 2 /4 |x| −2 or V (x) = (1/4) dist(x, ∂Ω) −2 , under Dirichlet boundary conditions. In this work we obtain sharp two-sided estimates on the corresponding heat kernels. To this end we transform the Schrödinger operators into suitable degenerate operators, for which we prove a new parabolic Harnack inequality up to the boundary. To derive the Harnack inequality we have established a series of new inequalities such as improved Hardy, logarithmic Hardy Sobolev, Hardy-Moser and weighted Poincaré. As a byproduct of our technique we are able to answer positively to a conjecture of E. B. Davies.

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“…The notion of the intrinsic ultracontractivity above was introduced in [11]. It is a very important concept in both analysis and probability, and has been studied extensively.…”

Section: Introductionmentioning

confidence: 99%

Intrinsic ultracontractivity for non-symmetric Lévy processes

Kim

Song²

2009

Forum Mathematicum

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Recently in [17,18], we extended the concept of intrinsic ultracontractivity to nonsymmetric semigroups and proved that for a large class of non-symmetric diffusions Z with measure-valued drift and potential, the semigroup of Z D (the process obtained by killing Z upon exiting D) in a bounded domain is intrinsic ultracontractive under very mild assumptions.In this paper, we study the intrinsic ultracontractivity for non-symmetric discontinuous Lévy processes. We prove that, for a large class of non-symmetric discontinuous Lévy processes X such that the Lebesgue measure is absolutely continuous with respect to the Lévy measure of X, the semigroup of X D in any bounded open set D is intrinsic ultracontractive. In particular, for the non-symmetric stable process X discussed in [24], the semigroup of X D is intrinsic ultracontractive for any bounded set D. Using the intrinsic ultracontractivity, we show that the parabolic boundary Harnack principle is true for those processes. Moreover, we get that the supremum of the expected conditional lifetimes in a bounded open set is finite. We also have results of the same nature when the Lévy measure is compactly supported.

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Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians

Cited by 370 publications

References 24 publications

On Talagrand's deviation inequalities for product measures

On Talagrand's deviation inequalities for product measures

Sharp Two–Sided Heat Kernel Estimates for Critical Schrödinger Operators on Bounded Domains

Intrinsic ultracontractivity for non-symmetric Lévy processes

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