Sharp Inequalities for Heat Kernels of Schrödinger Operators and Applications to Spectral Gaps

Bañuelos, Rodrigo; Méndez-Hernández, Pedro J.

doi:10.1006/jfan.2000.3611

Cited by 31 publications

(34 citation statements)

References 16 publications

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“…The full conjecture is only known in dimension one, see [28]. For zero potentials it has been proved in [9] and [20] for certain planar domains with symmetry. From the probabilistic point of view, the spectral gap determines the rate to equilibrium for Brownian motion conditioned to remain forever in D, the Doob h-process corresponding to the ground state eigenfunction.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Spectral Gap for the Cauchy Process on Convex, Symmetric Domains

Bañuelos

Kulczycki

2006

Communications in Partial Differential Equations

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Let D ⊂ R 2 be a bounded convex domain which is symmetric relative to both coordinate axes. Assume that [−a, a] × [−b, b], a ≥ b > 0 is the smallest rectangle (with sides parallel to the coordinate axes) containing D. Let {λ n } ∞ n=1 be the eigenvalues corresponding to the semigroup of the Cauchy process killed upon exiting D. We obtain the following estimate on the spectral gap:where C is an absolute constant. The estimate is obtained by proving new weighted Poincaré inequalities and appealing to the connection between the eigenvalue problem for the Cauchy process and a mixed boundary value problem for the Laplacian in one dimension higher known as the mixed Steklov problem established in [5].

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Spectral Gap for the Cauchy Process on Convex, Symmetric Domains

Bañuelos

Kulczycki

2006

Communications in Partial Differential Equations

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“…Then in Section 3 we state some applications of the main result to ratios of heat kernels and to ratios of integrals of heat kernels. These inequalities were first proved for a smaller class of domains by Bañuelos and Méndez-Hernández [5], and by You [25]. In Sections 4 and 5 we will give proofs of these applications.…”

Section: Introductionmentioning

confidence: 86%

Rearrangement Inequalities with Application to Ratios of Heat Kernels

Draghici

2005

Potential Anal

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We prove rearrangement inequalities for multiple integrals, using the polarization technique. Polarization refers to rearranging a function with respect to a hyperplane. Then we derive sharp inequalities for ratios of integrals of heat kernels of Schrödinger operators, using our polarization inequalities. These ratio inequalities imply inequalities for the partition functions and extend the results of R. Bañuelos, P.J. Méndez-Hernández and D. You. (2000): 26D15, 28A25, 35J10. Mathematics Subject Classifications

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“…Using martingale transforms the constant 2 is obtained in [3] and is reduced to 1.575 in [1]. Martingale techniques have been used subsequently to improve L p -bounds for the Grassmann operator, S, see [2].…”

Section: The Beurling Ahlfors Transformmentioning

confidence: 99%

Fourier Multipliers and Dirac Operators

Nolder

Wang

2017

Adv. Appl. Clifford Algebras

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Abstract. We use Fourier multipliers of the Dirac operator and Cauchy transform to obtain composition theorems and integral representations. In particular we calculate the multiplier of the Π-operator. This operator is the hypercomplex version of the Beurling Ahlfors transform in the plane. The hypercomplex Beuling Ahlfors transform is a direct generalization of the Beurling Ahlfors transform and reduces to this operator in the plane. We give an integral representation for iterations of the hypercomplex Beurling Ahlfors transform and we present here a bound for the L p -norm. Such L p -bounds are essential for applications of the Beurling Ahlfors transformation in the plane. The upper bound presented here is m(p * − 1) where m is the dimension of the Euclidean space on which the functions are defined, 1 < p < ∞ and p * = max(p, p/(p − 1)). We use recent estimates on second order Riesz transforms to obtain this result. Using the Fourier multiplier of the Π operator we express this operator as a hypercomplex linear combination of second order Riesz transforms.

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Sharp Inequalities for Heat Kernels of Schrödinger Operators and Applications to Spectral Gaps

Cited by 31 publications

References 16 publications

Spectral Gap for the Cauchy Process on Convex, Symmetric Domains

Spectral Gap for the Cauchy Process on Convex, Symmetric Domains

Rearrangement Inequalities with Application to Ratios of Heat Kernels

Fourier Multipliers and Dirac Operators

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