Unimodular Fourier multipliers for modulation spaces

By 'Arp'ad B'enyi, Karlheinz Gröchenig, Kasso Okoudjou, Luke G. Rogers. 2006

251 citations recorded by the source corpus.

Abstract We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol e i | ξ | α , where α ∈ [ 0 , 2 ] , are bounded on all modulation spaces, but, in general, fail to be bounded on the usual L p -spaces. As a consequence, the phase-space concentration of the solutions to the free Schrodinger and wave equations are preserved. As a byproduct, we also obtain boundedness results on modulation spaces for singular multipliers | ξ | − δ sin ( | ξ | α ) for 0 ⩽ δ ⩽ α .

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bf013fd8d34a9b280ed74d6100501bf52844a058

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Strichartz estimates in Wiener amalgam spaces for the Schrödinger equation (2006, methodology): “[1, 2]) For a ∈ R, a 6= 0, let fa(x) = (ai)e 2/(ai).”
On the global boundedness of Fourier integral operators (2008, background): “, for symbols σ(x, η) of order zero, see also [2, 4, 5].”
Changes of variables in modulation and Wiener amalgam spaces (2008, methodology): “2 that propagators of the form eit|D | α are actually locally continuous on M (R ) and W (R ) for all p, q and all t, α ∈ R (compared to the case of 0 ≤ α ≤ 2 on M (R ) analysed in [3] and to the well-known loss of derivatives in local L spaces, e.”
Sharpness of some properties of Wiener amalgam and modulation spaces (2008, unknown): “As an example, in Section 4 we prove the optimality of an estimate for the Schrödinger propagator, recently obtained in [22] (see also [1]).”
Local well‐posedness of nonlinear dispersive equations on modulation spaces (2007, background): “Thus, for 1 ≤ p ≤ ∞, and 0 < q ≤ ∞ the result follows from [3] and Lemma 2.”
Functions operating on modulation spaces and nonlinear dispersive equations (2014, methodology): “This is a classical theorem of Hörmander [12, 15], where as all the multipliers e a , 0 ≤ a ≤ 2 defines bounded Fourier multiplier operators on M s (R ) for 1 ≤ p, q ≤ ∞, s ≥ 0, as shown by Bényi-Gröcheing-Okoudjou-Rogers in [2], see also [1].”
The inclusion relation between Sobolev and modulation spaces (2010, background): “For example, the Schrödinger and wave propagators, which are not bounded on neither L nor B s , are bounded on M p,q s ([2]).”