Regularity properties of Schrödinger operators (original) (raw)
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1 Regularity Estimates in Hölder Spaces for Schrödinger
2016
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.
Regularity theory for the fractional harmonic oscillator
Journal of Functional Analysis, 2011
In this paper we develop the theory of Schauder estimates for the fractional harmonic oscillator H σ = (−∆ + |x| 2) σ , 0 < σ < 1. More precisely, a new class of smooth functions C k,α H is defined, in which we study the action of H σ. In fact these spaces are those adapted to the operator H, hence the suited ones for this type of regularity estimates. In order to prove our results, an analysis of the interaction of the Hermite-Riesz transforms with the Hölder spaces C k,α H is needed, that we believe of independent interest.
Regularity estimates in Hölder spaces for Schrödinger operators via a T1$$ theorem
Annali di Matematica Pura ed Applicata, 2012
We derive Hölder regularity estimates for operators associated with a timeindependent 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.
Global Hölder regularity for eigenfunctions of the fractional g-Laplacian
Journal of Mathematical Analysis and Applications
We establish global Hölder regularity for eigenfunctions of the fractional g−Laplacian with Dirichlet boundary conditions where g = G ′ and G is a Young functions satisfying the so called ∆2 condition. Our results apply to more general semilinear equations of the form (−∆g) s u = f (u).
Extension Problem and Harnack's Inequality for Some Fractional Operators
Communications in Partial Differential Equations, 2010
The fractional Laplacian can be obtained as a Dirichlet-to-Neumann map via an extension problem to the upper half space. In this paper we prove the same type of characterization for the fractional powers of second order partial differential operators in some class. We also get a Poisson formula and a system of Cauchy-Riemann equations for the extension. The method is applied to the fractional harmonic oscillator H σ = (−∆ + |x| 2) σ to deduce a Harnack's inequality. A pointwise formula for H σ f (x) and some maximum and comparison principles are derived.
Applied Mathematics, 2014
In this paper we consider operators with endpoint singularities generated by linear fractional Carleman shift in weighted Hölder spaces. Such operators play an important role in the study of algebras generated by the operators of singular integration and multiplication by function. For the considered operators, we obtained more precise relations between norms of integral operators with local singularities in weighted Lebesgue spaces and norms in weighted Hölder spaces, making use of previously obtained general results. We prove the boundedness of operators with linear fractional singularities.
The effect of the Hardy potential in some Calderón–Zygmund properties for the fractional Laplacian
Journal of Differential Equations, 2016
The goal of this paper is to study the effect of the Hardy potential on the existence and summability of solutions to a class of nonlocal elliptic problems (−∆) s u − λ u |x| 2s = f (x, u) in Ω, u = 0 in R N \ Ω, u > 0 in Ω, where (−∆) s , s ∈ (0, 1), is the fractional laplacian operator, Ω ⊂ R N is a bounded domain with Lipschitz boundary such that 0 ∈ Ω and N > 2s. We will mainly consider the solvability in two cases: (1) The linear problem, that is, f (x, t) = f (x), where according to the summability of the datum f and the parameter λ we give the summability of the solution u. (2) The problem with a nonlinear term f (x, t) = h(x) t σ for t > 0. In this case, existence and regularity will depend on the value of σ and on the summability of h. Looking for optimal results we will need a weak Harnack inequality for elliptic operators with singular coefficients that seems to be new.
A Characterization of Hardy Spaces Associated with Certain Schrödinger Operators
Potential Analysis, 2014
Let {K t } t>0 be the semigroup of linear operators generated by a Schrödinger operator −L = Δ − V (x) on R d , d ≥ 3, where V (x) ≥ 0 satisfies Δ −1 V ∈ L ∞. We say that an L 1-function f belongs to the Hardy space H 1 L if the maximal function M L f (x) = sup t>0 |K t f (x)| belongs to L 1 (R d). We prove that the operator (−Δ) 1/2 L −1/2 is an isomorphism of the space H 1 L with the classical Hardy space H 1 (R d) whose inverse is L 1/2 (−Δ) −1/2. As a corollary we obtain that the space H 1 L is characterized by the Riesz transforms R j = ∂ ∂x j L −1/2 .