Unimodular Fourier multipliers for modulation spaces

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 Schrödinger 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.

Estimates for unimodular Fourier multipliers on modulation spaces

“…They arise when solving the Cauchy problem for dispersive equations. For example, for the solution u(t, x) of the Cauchy problem (1) i∂ t u + |∆| α/2 u = 0, [18,20]). It is then natural to study boundedness properties on other function spaces arising in Fourier analysis.…”

“…It is then natural to study boundedness properties on other function spaces arising in Fourier analysis. This was recently done in [1] for the modulation spaces M p,q , 1 ≤ p, q ≤ ∞. These spaces were first introduced by Feichtinger [10,11] to measure smoothness of a function or distribution in a way different from Besov spaces, and they are now recognized as a useful tool for studying pseudodifferential operators (see [14,22,25] [15,25,28]).…”

Strichartz estimates in Wiener amalgam spaces for the Schrödinger equation

“…In other terms, estimate (7) contains information on the oscillations in x of the solution, which cannot be extracted from (2).…”

Modulation space estimates for Schrödinger type equations with time-dependent potentials

“…As mentioned before, the unimodular Fourier multiplier is not a bounded operator on any L q in general except for q = 2. However, a recent work by Bényi-Gröchenig-Okoudjou-Rogers [1] has shown that e −it(−∆) κ/2 with κ ∈ [0, 2] preserves the M p,qnorm for any 1 p, q ∞, which is proved by the stationary phase method. More precisely, we have the estimate…”