Wave front set at infinity for tempered ultradistributions and hyperbolic equations

Cappiello, Marco

doi:10.1007/s11565-006-0019-0

Ann. Univ. Ferrara

2006

DOI: 10.1007/s11565-006-0019-0

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Wave front set at infinity for tempered ultradistributions and hyperbolic equations

Marco Cappiello

Abstract: We study the propagation of singularities and the microlocal behaviour at infinity for the solution of the Cauchy problem associated to an SG-hyperbolic operator with one characteristic of constant multiplicity. We perform our analysis in the framework of tempered ultradistributions, cf. Introduction, using an appropriate notion of wave front set.

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2020

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The Gabor Wave Front Set of Compactly Supported Distributions

Wahlberg

2020

Applied and Numerical Harmonic Analysis

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We show that the Gabor wave front set of a compactly supported distribution equals zero times the projection on the second variable of the classical wave front set.Dedicated to Luigi Rodino on the occasion of his 70 th birthday IntroductionThe Gabor wave front set of tempered distributions was introduced by Hörmander 1991 [8]. The idea was to measure singularities of tempered distributions both in terms of smoothness and decay at infinity comprehensively. Using the short-time Fourier transform the Gabor wave front set can be described as the directions in phase spaceThe Gabor wave front set behaves differently than the classical C ∞ wave front set, also introduced by Hörmander 1971 (see [7, Chapter 8]). It is for example translation invariant. The Gabor wave front set is adapted to the Shubin calculus of pseudodifferential operators [16] where symbols have isotropic behavior in phase space. With respect to this calculus, the corresponding notion of characteristic set, and the Gabor wave front set, pseudodifferential operators are microlocal and microelliptic, similar to pseudodifferential operators with Hörmander symbols in their natural context.The main result of this note concerns the Gabor wave front set of compactly supported distributions. We show (see Corollary 3.4) that for u ∈ E ′ (R d ) we havewhere WF G denotes the Gabor wave front set, WF denotes the classical wave front set, and π 2 is the projection on the covariable (second) phase space R d coordinate. By [7, Theorem 8.1.3], π 2 WF(u) = Σ(u). The symbol Σ(u) denotes the cone complement of the space of directions in R d in an open conic neighborhood of which the Fourier transform of u ∈ E ′ (R d ) decays rapidly. The equality (1.1) thus describes exactly the Gabor wave front set of u ∈ E ′ (R d ) in terms of known ingredients in terms of WF(u): The space coordinate is zero and the frequency directions are exactly the "irregular" frequency directions of u.In the literature there are several concepts of global wave front sets apart from the Gabor wave front set. There is a parametrized version [17] and an Gelfand-Shilov version [2] of the same idea. Melrose [12] introduced the scattering wave front set which 2010 Mathematics Subject Classification. 35A18, 35A21, 46F05, 46F12.

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The Gabor Wave Front Set of Compactly Supported Distributions

Wahlberg

2020

Applied and Numerical Harmonic Analysis

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Wave front set at infinity for tempered ultradistributions and hyperbolic equations

Cited by 1 publication

References 22 publications

The Gabor Wave Front Set of Compactly Supported Distributions

The Gabor Wave Front Set of Compactly Supported Distributions

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