Fourier restriction estimates on quantum Euclidean spaces

Effacement des dérivations et spectres premiers des algèbres quantiques

Quantum Groups and Integrable Models

We generalize the theorems of Stein--Tomas and Strichartz about surface\nrestrictions of Fourier transforms to systems of orthonormal functions with an\noptimal dependence on the number of functions. We deduce the corresponding\nStrichartz bounds for solutions to Schr\\"odinger equations up to the endpoint,\nthereby solving an open problem of Frank, Lewin, Lieb and Seiringer. We also\nprove uniform Sobolev estimates in Schatten spaces, extending the results of\nKenig, Ruiz, and Sogge. We finally provide applications of these results to a\nLimiting Absorption Principle in Schatten spaces, to the well-posedness of the\nHartree equation in Schatten spaces, to Lieb--Thirring bounds for eigenvalues\nof Schr\\"odinger operators with complex potentials, and to Schatten properties\nof the scattering matrix.\n

We establish variational estimates related to the problem of restricting the Fourier transform of a three-dimensional function to the two-dimensional Euclidean sphere. At the same time, we give a short survey of the recent field of maximal Fourier restriction theory.

We extend the mapping properties for the fractional integral operators, the convolution operators, the Fourier integral operators and the oscillatory integral operators to rearrangementinvariant quasi-Banach function spaces.We also generalize the Fourier restriction theorem and the Sobolev embedding theorem to rearrangement-invariant quasi-Banach function spaces.We obtain the above results by introducing two families of rearrangement-invariant quasi-Banach function spaces.Furthermore, these two families of rearrangement-invariant quasi-Banach function spaces also give us some embedding and interpolation results of Triebel-Lizorkin type spaces and Hardy type spaces built on rearrangement-invariant quasi-Banach function spaces.

This chapter mostly considers the domains of type Dsubscript (l), which are in some sense “closest” to the principal root jet, since it turns out that the other domains Dsubscript (l) with l ≥ 2 are easier to handle. In a first step, by means of some lower bounds on the r-height, this chapter establishes favorable restriction estimates in most situations, with the exception of certain cases where m = 2 and B = 3 or B = 4. In some cases the chapter applies interpolation arguments in order to capture the endpoint estimates for p = psubscript c. Sometimes this can be achieved by means of a variant of the Fourier restriction theorem. However, in most of these cases the chapter applies complex interpolation in a similar way as has been done in Chapter 5.

We present a restriction theorem for the Fourier transform to a 2-dimensional conical surface of finite type, obtaining a sharp result, which improves previous work by Barcelo.

A sharp $$L^p-L^q$$ L p - L q -Fourier restriction theorem for a conical surface of finite type