Kernel theorems for the spaces of tempered ultradistributions

Abstract. We consider a class of linear Schrödinger equations in R d , with analytic symbols. We prove a global-in-time integral representation for the corresponding propagator as a generalized Gabor multiplier with a window analytic and decaying exponentially at infinity, which is transported by the Hamiltonian flow. We then provide three applications of the above result: the exponential sparsity in phase space of the corresponding propagator with respect to Gabor wave packets, a wave packet characterization of Fourier integral operators with analytic phases and symbols, and the propagation of analytic singularities.

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“…To present the result in full generality, we recall the following Kernel Theorem for ultra-distributions [34].…”

Section: Representation As Classical Fiomentioning

confidence: 99%

Wave packet analysis of Schrödinger equations in analytic function spaces

Cordero

Nicola

Rodino

2015

Advances in Mathematics

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“…We need the following kernel theorem for S ′ * from [16]. The (M p ) case was already considered in [9] (the authors used the characterization of Fourier-Hermite coefficients of the elements of the space in the proof of the kernel theorem). Proposition 1.2.…”

Section: Preliminariesmentioning

confidence: 99%

Anti-Wick and Weyl quantization on ultradistribution spaces

Pilipović

Prangoski

2015

Journal de Mathématiques Pures et Appliquées

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The connection between the Anti-Wick and Weyl quantization is given for certain class of global symbols, which corresponding pseudodifferential operators act continuously on the space of tempered ultradistributions of Beurling, respectively, of Roumieu type. The largest subspace of ultradistributions is found for which the convolution with the gaussian kernel exist. This gives a way to extend the definition of Anti-Wick quantization for symbols that are not necessarily tempered ultradistributions.The spaces of ultradistributions and ultradistributions with compact support of Beurling and Roumieu type are defined as the strong duals of D (Mp) (U) and E (Mp) (U), resp. D {Mp} (U) and E {Mp} (U). For the properties of these spaces, we refer to [6], [7] and [8]. In the future we will not emphasize the set U when U = R d . Also, the common notation for the symbols (M p ) and {M p } will be *.For f ∈ L 1 , its Fourier transform is defined byBy R is denoted a set of positive sequences which monotonically increases to infinity. For (r p ) ∈ R, consider the sequence N 0 = 1, N p = M p p j=1 r j , p ∈ Z + . One easily sees that this sequence satisfies (M.1) and (M.3) ′ and its associated function will be denoted by N rp (ρ), i.e. N rp (ρ) = sup p∈N log + ρ p M p p j=1 r j , ρ > 0. Note, for given (r p ) and every k > 0 there is ρ 0 > 0 such that N rp (ρ) ≤ M(kρ), for ρ > ρ 0 . In [8] it is proven that for each K ⊂⊂ R d , the topology of D {Mp} K = lim −→ h→∞ D {Mp},h K is generated by the seminorms p (t j ),K (ϕ) = sup α∈N d D α ϕ L ∞

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“…Since we are especially interested in the behaviour of corrsponding Schwartz kernels, the following result is important to us. The proof is omited since it can be found in [5]. to D ′ s (Ω 2 ), then there is a unique ultradistribution…”

Section: Preliminariesmentioning

confidence: 99%

Boundedness of Gevrey and Gelfand–Shilov kernels of positive semi-definite operators

Chen

Toft

2015

J. Pseudo-Differ. Oper. Appl.

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We study properties of positive operators on Gelfand-Shilov spaces, and distributions which are positive with respect to non-commutative convolutions. We prove that boundedness of kernels K ∈ D ′ s to positive operators, are completely determined by the behaviour of K alone the diagonal. We also prove that positive elements a in S ′ with respect to twisted convolutions, having Gevrey class property of order s ≥ 1/2 at the origin, then a belongs to the Gelfand-Shilov space S s .

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Kernel theorems for the spaces of tempered ultradistributions

Cited by 11 publications

References 19 publications

Wave packet analysis of Schrödinger equations in analytic function spaces

Wave packet analysis of Schrödinger equations in analytic function spaces

Anti-Wick and Weyl quantization on ultradistribution spaces

Boundedness of Gevrey and Gelfand–Shilov kernels of positive semi-definite operators

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