Highly oscillatory unimodular Fourier multipliers on modulation spaces

We study the continuity on the modulation spaces \(M^{p,q}\) of Fourier multipliers with symbols of the type \(e^{i\mu(\xi)}\), for some real-valued function \(\mu(\xi)\). A number of results are known, assuming that the derivatives of order \(\geq 2\) of the phase \(\mu(\xi)\) are bounded or, more...

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Bibliographic Details
Published inarXiv.org
Main Authors Nicola, Fabio, Primo, Eva, Tabacco, Anita
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 19.01.2018
Subjects
Derivatives
Mathematical functions
Modulation
Multipliers
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Summary:We study the continuity on the modulation spaces \(M^{p,q}\) of Fourier multipliers with symbols of the type \(e^{i\mu(\xi)}\), for some real-valued function \(\mu(\xi)\). A number of results are known, assuming that the derivatives of order \(\geq 2\) of the phase \(\mu(\xi)\) are bounded or, more generally, that its second derivatives belong to the Sj\"ostrand class \(M^{\infty,1}\). Here we extend those results, by assuming that the second derivatives lie in the bigger Wiener amalgam space \(W(\mathcal{F} L^1,L^\infty)\); in particular they could have stronger oscillations at infinity such as \(\cos |\xi|^2\). Actually our main result deals with the more general case of possibly unbounded second derivatives. In that case we have boundedness on weighted modulation spaces with a sharp loss of derivatives.
ISSN:2331-8422