Fourier Restriction Research Articles

Well-posedness for the NLS hierarchy

We prove well-posedness for higher-order equations in the so-called NLS hierarchy (also known as part of the AKNS hierarchy) in almost critical Fourier–Lebesgue spaces and in modulation spaces. We show the jth equation in the hierarchy is locally well-posed for initial data in H^rs(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\hat{H}}}^s_r({\\mathbb {R}})$$\\end{document} for s≥j-1r′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s \\ge \\frac{j-1}{r'}$$\\end{document} and 1<r≤2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 < r \\le 2$$\\end{document} and also in M2,ps(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M^s_{2, p}({\\mathbb {R}})$$\\end{document} for s=j-12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s = \\frac{j-1}{2}$$\\end{document} and 2≤p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2 \\le p < \\infty $$\\end{document}. Supplementing our results with corresponding ill-posedness results in Fourier–Lebesgue and modulation spaces shows optimality. Using the conserved quantities derived in Koch and Tataru (Duke Math J 167(17), 3207–3313, 2018), we argue that the hierarchy equations are globally well-posed for data in Hs(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^s({\\mathbb {R}})$$\\end{document} for s≥j-12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s \\ge \\frac{j-1}{2}$$\\end{document}. Our arguments are based on the Fourier restriction norm method in Bourgain spaces adapted to our data spaces and bi- and trilinear refinements of Strichartz estimates.

Tomographic Fourier extension identities for submanifolds of Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document}

We establish identities for the composition Tk,n(|gdσ^|2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{k,n}(|\\widehat{gd\\sigma }|^2)$$\\end{document}, where g↦gdσ^\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g\\mapsto \\widehat{gd\\sigma }$$\\end{document} is the Fourier extension operator associated with a general smooth k-dimensional submanifold of Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document}, and Tk,n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T_{k,n}$$\\end{document} is the k-plane transform. Several connections to problems in Fourier restriction theory are presented.

Pointwise convergence problem of the one‐dimensional Boussinesq‐type equation

In this paper, we consider the initial data problem of the one‐dimensional Boussinesq‐type equation. Inspired by the work of Compaan et al. (Int. Math. Res. Not. 2021, 599–650), by using the Fourier restriction norm method, we investigate the pointwise convergence of the one‐dimensional Boussinesq equation and cubic Boussinesq equation. We also only use the Strichartz estimates instead of Fourier restriction norm method to establish convergence conclusions of the one‐dimensional cubic Boussinesq equation. The key ingredients are some Strichartz estimates and high‐low frequency decomposition related to the nonlinear part of the integral form solution in the Duhamel form and the maximal function estimate related to inhomogeneous estimate.

Existence of Extremals for a Fourier Restriction Inequality on the One-Sheeted Hyperboloid

Existence of Extremals for a Fourier Restriction Inequality on the One-Sheeted Hyperboloid

Well-posedness of the periodic dispersion-generalized Benjamin–Ono equation in the weakly dispersive regime

We study the dispersion-generalized Benjamin–Ono equation in the periodic setting. This equation interpolates between the Benjamin–Ono equation ( α=1 ) and the inviscid Burgers’ equation ( α=0 ). We obtain local well-posedness in Hs(T) for s>32−α and α∈(0,1) by using the short-time Fourier restriction method.

The endpoint Stein–Tomas inequality: old and new

AbstractThe Stein–Tomas inequality from 1975 is a cornerstone of Fourier restriction theory. Despite its respectable age, it is a fertile ground for current research. This note is centered around three classical applications – to Strichartz inequalities, Salem sets and Roth’s theorem in the primes – and three recent improvements: the sharp endpoint Stein–Tomas inequality in three space dimensions, maximal and variational refinements, and the symmetric Stein–Tomas inequality with applications.

The cubic nonlinear fractional Schrödinger equation on the half-line

We study the cubic nonlinear fractional Schrödinger equation with Lévy indices 43<α<2 posed on the half-line. More precisely, we define the notion of a solution for this model and we obtain a result of local-well-posedness almost sharp in the sense of index of regularity required for the solutions with respect for known results on the full real line R. Also, we prove for the same model that the solution of the nonlinear part is smoother than the initial data and the corresponding linear solution. To get our results we use the Colliander and Kenig approach based on the Riemann–Liouville fractional operator combined with Fourier restriction method of Bourgain (1993) and some ideas of the recent work of Erdoğan et al. (2019). The method applies to both focusing and defocusing nonlinearities. As a consequence of our analysis we prove a smoothing effect for the cubic nonlinear fractional Schrödinger equation posed in full line R for the case of the low regularity assumption, which was point out at the recent work (Erdoğan et al., 2019).

Uniform maximal Fourier restriction for convex curves

Uniform maximal Fourier restriction for convex curves

A Family of Fractal Fourier Restriction Estimates with Implications on the Kakeya Problem

A Family of Fractal Fourier Restriction Estimates with Implications on the Kakeya Problem

Sharp Fourier extension on the circle under arithmetic constraints

We establish a sharp adjoint Fourier restriction inequality for the end-point Tomas-Stein restriction theorem on the circle under a certain arithmetic constraint on the support set of the Fourier coefficients of the given function. Such arithmetic constraint is a generalization of a B3-set.

Decoupling Inequality for Paraboloid Under Shell Type Restriction and Its Application to the Periodic Zakharov System

In this paper, we establish local well-posedness for the Zakharov system on {mathbb {T}}^d, dge 3 in a low regularity setting. Our result improves the work of Kishimoto (J Anal Math 119:213–253, 2013). Moreover, the result is sharp up to varepsilon -loss of regularity when d=3 and dge 5 as long as one utilizes the iteration argument. We introduce ideas from recent developments of the Fourier restriction theory. The key element in the proof of our well-posedness result is a new trilinear discrete Fourier restriction estimate involving paraboloid and cone. We prove this trilinear estimate by improving Bourgain–Demeter’s range of exponent for the linear decoupling inequality for paraboloid (Bourgain and Demeter in Ann Math 182:351–389) under the constraint that the input space-time function f satisfies textrm{supp}, {hat{f}} subset { (xi ,tau ) in {mathbb {R}}^{d+1}: 1- frac{1}{N} le |xi | le 1 + frac{1}{N},; |tau - |xi |^2| le frac{1}{N^{2}} } for large Nge 1.

Fourier restriction estimates on quantum Euclidean spaces

In this paper, we initiate the study of the Fourier restriction phenomena on quantum Euclidean spaces, and establish the analogues of the Tomas-Stein restriction theorem and the two-dimensional full restriction theorem.

A note on Fourier restriction and nested Polynomial Wolff axioms

A note on Fourier restriction and nested Polynomial Wolff axioms

Factorization in Fourier restriction theory and near extremizers

AbstractIn Fourier restriction theory, abstract so‐called Maurey–Nikishin–Pisier factorization has often been used as a convenient off‐the‐shelf means to prove certain factorizations of the Fourier restriction operator. We give an alternative approach to such factorizations in Fourier restriction theory. Based on an induction‐on‐scales argument, our comparably simple method applies to any compact quadratic surface, in particular compact parts of the paraboloid and the hyperbolic paraboloid. This is achieved by constructing near extremizers with big “mass,” which itself might be of interest.

The Stein–Tomas inequality under the effect of symmetries

We prove new Fourier restriction estimates to the unit sphere {mathbb{S}^{d-1}} on the class of O(d − k) × O(k)-symmetric functions, for every d ≥ 4 and 2 ≤ k ≤ d − 2. As an application, we establish the existence of maximizers for the endpoint Stein–Tomas inequality within that class. Moreover, we construct examples showing that the range of Lebesgue exponents in our estimates is sharp.

Real Analysis, Harmonic Analysis and Applications

The workshop focused on important developments within the last few years in real and harmonic analysis including nonlinear Carleson theorems and singular integral theory, Fourier restriction theory and spherical maximal functions as well as concurrent progress in the application of these for example to partial differential equations.

Spectral projectors, resolvent, and Fourier restriction on the hyperbolic space

We develop a unified approach to proving Lp−Lq boundedness of spectral projectors, the resolvent of the Laplace-Beltrami operator and its derivative on Hd. In the case of spectral projectors, and when p and q are in duality, the dependence of the implicit constant on p is shown to be sharp. We also give partial results on the question of Lp−Lq boundedness of the Fourier extension operator. As an application, we prove smoothing estimates for the free Schrödinger equation on Hd and a limiting absorption principle for the electromagnetic Schrödinger equation with small potentials.

Existence of extremizers for Fourier restriction to the moment curve

We show that the restriction and extension operators associated to the moment curve possess extremizers and that L p L^p -normalized extremizing sequences of these operators are precompact modulo symmetries.

Low regularity a priori estimate for KDNLS via the short-time Fourier restriction method

In this article, we consider the kinetic derivative nonlinear Schrödinger equation (KDNLS), which is a one-dimensional nonlinear Schrödinger equation with a cubic derivative nonlinear term containing the Hilbert transformation. For the Cauchy problem, both on the real line and on the circle, we apply the short-time Fourier restriction method to establish a priori estimate for small and smooth solutions in Sobolev spaces H^{s} with s>1/4.

Decoupling inequalities for quadratic forms

We prove sharp $\ell^q L^p$ decoupling inequalities for $p,q \in [2,\infty)$ and arbitrary tuples of quadratic forms. Connections to prior results on decoupling inequalities for quadratic forms are also explained. We also include some applications of our results to exponential sum estimates and to Fourier restriction estimates. The proof of our main result is based on scale-dependent Brascamp--Lieb inequalities.