Regularity properties of Schrödinger operators

Ma, Tao; Stinga, P. R.; Torrea, José L.; Zhang, Chao

doi:10.1016/j.jmaa.2011.10.006

Cited by 42 publications

(78 citation statements)

References 16 publications

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“…This same result, only for the case when α + 2s is not an integer, was obtained using pointwise formulas and the Schauder estimates for the Laplacian in [84,85]. In our case the semigroup method permits us to include the case when α + 2s ∈ N. See [29,38,50,69,76,92] for similar proofs in different contexts.…”

Section: Theorem 12 (Hölder Estimatessupporting

confidence: 67%

User’s guide to the fractional Laplacian and the method of semigroups

Stinga¹

2019

Fractional Differential Equations

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The method of semigroups is a unifying, widely applicable, general technique to formulate and analyze fundamental aspects of fractional powers of operators L and their regularity properties in related functional spaces. The approach was introduced by the author and José L. Torrea in 2009 (arXiv:0910.2569v1). The aim of this chapter is to show how the method works in the particular case of the fractional Laplacian L s = (−∆) s , 0 < s < 1. The starting point is the semigroup formula for the fractional Laplacian. From here, the classical heat kernel permits us to obtain the pointwise formula for (−∆) s u(x). One of the key advantages is that our technique relies on the use of heat kernels, which allows for applications in settings where the Fourier transform is not the most suitable tool. In addition, it provides explicit constants that are key to prove, under minimal conditions on u, the validity of the pointwise limitsThe formula for the solution to the Poisson problem (−∆) s u = f is found through the semigroup approach as the inverse of the fractional Laplacian u(x) = (−∆) −s f (x) (fundamental solution). We then present the Caffarelli-Silvestre extension problem, whose explicit solution is given by the semigroup formulas that were first discovered by the author and Torrea. With the extension technique, an interior Harnack inequality and derivative estimates for fractional harmonic functions can be obtained. The classical Hölder and Schauder estimates (−∆) ±s : C α → C α∓2s are proved with the method of semigroups in a rather quick, elegant way. The crucial point for this will be the characterization of Hölder and Zygmund spaces with heat semigroups.

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Section: Theorem 12 (Hölder Estimatessupporting

confidence: 67%

User’s guide to the fractional Laplacian and the method of semigroups

Stinga¹

2019

Fractional Differential Equations

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“…As already mentioned in the paper by Bongioanni, Harboure and Salinas [6], the BM O α L spaces are the duals of the H p L spaces defined in [12,13,14]. In fact, if s > n and 0 [15], references in [20] and [26].…”

Section: The Spacesmentioning

confidence: 84%

Regularity estimates in Hölder spaces for Schrödinger operators via a $$T1$$ theorem

Stinga

Torrea

et al. 2012

Annali di Matematica

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Abstract. We derive Hölder regularity estimates for operators associated with a time independent Schrödinger operator of the form −∆ + V . The results are obtained by checking a certain condition on the function T 1. Our general method applies to get regularity estimates for maximal operators and square functions of the heat and Poisson semigroups, for Laplace transform type multipliers and also for Riesz transforms and negative powers (−∆ + V ) −γ/2 , all of them in a unified way.

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“…2), we can follow the proof of (d) of Proposition 3.6, [10] to show that for V ∈ B q , q ≥ n/2, there is some δ ∈ (0, 1) and N > max{4, 2 log(C 0 + 2 − n)} with C 0 the doubling constant of with respect to the potential V as in (1.1), such that for t > 0 and x ∈ R n ,…”

Section: )mentioning

confidence: 85%

“…We will use two facts about the space BMO L (R n )(see [4,5,10]) under the assumption of V ∈ B q , q ≥ n:…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

Carleson measures, BMO spaces and balayages associated to Schrödinger operators

Chen

Duong

et al. 2017

Sci. China Math.

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Abstract. Let L be a Schrödinger operator of the form L = −∆+V acting on L 2 (R n ), n ≥ 3, where the nonnegative potential V belongs to the reverse Hölder class B q for some q ≥ n. Let BMO L (R n ) denote the BMO space associated to the Schrödinger operator L on R n . In this article we show that for every f ∈ BMO L (R n ) with compact support, then there exist g ∈ L ∞ (R n ) and a finite Carleson measure µ such thatand P t (x, y) is the kernel of the Poisson semigroup {eConversely, if µ is a Carleson measure, then S µ,P belongs to the space BMO L (R n ). This extends the result for the classical John-Nirenberg BMO space by Carleson [1] (see also [13,7,14]) to the BMO setting associated to Schrödinger operators.

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Regularity properties of Schrödinger operators

Cited by 42 publications

References 16 publications

User’s guide to the fractional Laplacian and the method of semigroups

User’s guide to the fractional Laplacian and the method of semigroups

Regularity estimates in Hölder spaces for Schrödinger operators via a $$T1$$ theorem

Carleson measures, BMO spaces and balayages associated to Schrödinger operators

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