Estimates for a certain bilinear Fourier integral operator (original) (raw)

FA ] 2 4 A pr 2 00 8 ON THE GLOBAL BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS

2021

We consider a class of Fourier integral operators, globally defined on R, with symbols and phases satisfying product type estimates (the so-called SG or scattering classes). We prove a sharp continuity result for such operators when acting on the modulation spaces M. The minimal loss of derivatives is shown to be d|1/2−1/p|. This global perspective produces a loss of decay as well, given by the same order. Strictly related, striking examples of unboundedness on L spaces are presented.

Some new two-sided inequalities concerning the Fourier transform

Mathematical Inequalities & Applications, 2017

The classical Hausdorff-Young and Hardy-Littlewood-Stein inequalities do not hold for p > 2. In this paper we prove that if we restrict to net spaces we can even derive a two-sided estimate for all p > 1. In particular, this result generalizes a recent result by Liflyand E.

On the global boundedness of Fourier integral operators

Annals of Global Analysis and Geometry, 2010

We consider a class of Fourier integral operators, globally defined on R d , with symbols and phases satisfying product type estimates (the so-called SG or scattering classes). We prove a sharp continuity result for such operators when acting on the modulation spaces M p . The minimal loss of derivatives is shown to be d|1/2−1/p|. This global perspective produces a loss of decay as well, given by the same order. Strictly related, striking examples of unboundedness on L p spaces are presented.

Functions with Bounded Spectrum

Transactions of the American Mathematical Society, 1995

Let 0 < p < oo, f(x) e Lp(U.n), and supp Ff be bounded, where F is the Fourier transform. We will prove in this paper that the sequence ll-DQ/]|i/|a'; a > o , has the same behavior as the sequence sup |£Q|'/I<«I; {esuppf/ a > 0. In other words, if we know all "far points" of supp Ff, we can wholly describe this behavior without any concrete calculation of ||Z)a/||p , a > 0. A Paley-Wiener-Schwartz theorem for a nonconvex case, which is a consequence of the result, is given.