Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation

Abstract

We study the action of metaplectic operators on Wiener amalgam spaces, giving upper bounds for their norms. As an application, we obtain new fixed-time estimates in these spaces for Schrödinger equations with general quadratic Hamiltonians and Strichartz estimates for the Schrödinger equation with potentials V (x) = ±|x| 2 .

On the global boundedness of Fourier integral operators

“…There it was shown that A is bounded as an operator (F L p ) comp → (F L p ) loc , 1 ≤ p ≤ ∞, if m ≤ −d 1 2 − 1 p . This is similar to (5), but with the difference of one unit in the dimension. Surprisingly, this threshold was shown to be sharp in any dimension d ≥ 1, even for phases linear with respect to η; see [7] (or Sect.…”

Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces

“…In fact, there are some recent works which have been devoted to the study of the well-posedness for a class of nonlinear evolution equation in modulation spaces; cf. [2,3,5,7,8,9,20,24,25,26,31,39,40,41,42]. Our main goal of this paper is to study the global well-posedness of 4NLS in modulation spaces M 3+1/2 2,1…”

Boundedness of Fourier Integral Operators on $\mathcal {F}L^p$ spaces

“…Similarly, in [23], the main point was a decomposition of T into a part with phase which is nondegenerate with respect to η (and further factorized) and a degenerate part, with a phase closer to the linear case, fulfilling better estimates. However, as example (5) shows, the case of linear phases in η already contains all the obstructions to the local FL p -boundedness, so that we cannot here take advantage of this kind of decomposition. Instead, the proof of Theorem 1.1 makes use of the theory of modulation spaces M p , 1 ≤ p ≤ ∞, which are now classical function spaces used in time-frequency analysis (see [8,9,12] and Section 2 for the definition and properties).…”