Smoothing estimates for non-dispersive equations

Ruzhansky, Michael; Sugimoto, M.

doi:10.1007/s00208-015-1281-1

Cited by 10 publications

(13 citation statements)

References 44 publications

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“…If insted of the supersmoothing property we restrict to the weaker smoothing property, several additional results are available ( [23], [22] among the others). We mention in particular the following one, which was proved in [1], [3] and will be used below: …”

Section: Applicationsmentioning

confidence: 99%

Kato Smoothing and Strichartz Estimates for Wave Equations with Magnetic Potentials

D’Ancona

2014

Commun. Math. Phys.

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Abstract. Let H be a selfadjoint operator and A a closed operator on a Hilbert space H. If A is H-(super)smooth in the sense of Kato-Yajima, wesmooth. This allows to include wave and Klein-Gordon equations in the abstract theory at the same level of generality as Schrödinger equations.We give a few applications and in particular, based on the resolvent estimates of Erdogan, Goldberg and Schlag [9], we prove Strichartz estimates for wave equations perturbed with large magnetic potentials on R n , n ≥ 3.

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Section: Applicationsmentioning

confidence: 99%

Kato Smoothing and Strichartz Estimates for Wave Equations with Magnetic Potentials

D’Ancona

2014

Commun. Math. Phys.

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“…Now we consider what happens if the equation does not satisfy the dispersiveness assumption ∇a(ξ) = 0 (ξ ∈ R n ). All the precise results and arguments in this section are to appear in our forthcoming paper [RS2].…”

Section: Non-dispersive Casementioning

confidence: 99%

Recent Progress in Smoothing Estimates for Evolution Equations

Ruzhansky

Sugimoto

2013

Springer Proceedings in Mathematics &Amp; Statistics

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“…The general methodology of "comparison principles" was introduced in [22,24], where its usefulness was demonstrated by a wide array of operators.…”

Section: Comparison Principlesmentioning

confidence: 99%

Spectral identities and smoothing estimates for evolution operators

Ben‐Artzi¹,

Ruzhansky²,

Sugimoto³

2018

Preprint

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Smoothing (and decay) spacetime estimates are discussed for evolution groups of self-adjoint operators in an abstract setting. The basic assumption is the existence (and weak continuity) of the spectral density in a functional setting. Spectral identities for the time evolution of such operators are derived, enabling results concerning "best constants" for smoothing estimates. When combined with suitable "comparison principles" (analogous to those established in [22]), they yield smoothing estimates for classes of functions of the operators .A important particular case is the derivation of global spacetime estimates for a perturbed operator H + V on the basis of its comparison with the unperturbed operator H.A number of applications are given, including smoothing estimates for fractional Laplacians, Stark Hamiltonians and Schrödinger operators with potentials. Contents1. Introduction 2 2. The basic setup and notation 3 3. Global smoothing estimates 4 4. Comparison principles 9 4.1. Comparing unperturbed and perturbed operators 11 5. Applications to operators of mathematical physics 13 5.1. The fractional Laplacian 13 5.2. The Stark Hamiltonian 13 5.3. The Schrödinger operator with potential 14 References 15

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Smoothing estimates for non-dispersive equations

Cited by 10 publications

References 44 publications

Kato Smoothing and Strichartz Estimates for Wave Equations with Magnetic Potentials

Kato Smoothing and Strichartz Estimates for Wave Equations with Magnetic Potentials

Recent Progress in Smoothing Estimates for Evolution Equations

Spectral identities and smoothing estimates for evolution operators

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