scholarly journals Existence of extremals for a Fourier restriction inequality

2012 ◽  
Vol 5 (2) ◽  
pp. 261-312 ◽  
Author(s):  
Francis Michael Christ ◽  
Shuanglin Shao
Keyword(s):  
Fourier Restriction ◽  
Restriction Inequality

scholarly journals Global maximizers for the sphere adjoint Fourier restriction inequality

2015 ◽  
Vol 268 (3) ◽  
pp. 690-702 ◽  
Author(s):  
Damiano Foschi
Keyword(s):  
Fourier Restriction ◽  
Restriction Inequality

scholarly journals On the extremizers of an adjoint Fourier restriction inequality

2012 ◽  
Vol 230 (3) ◽  
pp. 957-977 ◽  
Author(s):  
Michael Christ ◽  
Shuanglin Shao
Keyword(s):  
Fourier Restriction ◽  
Restriction Inequality

scholarly journals Smoothness of solutions of a convolution equation of restricted type on the sphere

2021 ◽  
Vol 9 ◽  
Author(s):  
Diogo Oliveira e Silva ◽  
René Quilodrán
Keyword(s):  
A Priori ◽  
Companion Paper ◽  
Convolution Equation ◽  
Regularity Of Solutions ◽  
Surface Measure ◽  
Fourier Restriction ◽  
Sharp Form ◽  
Restricted Type ◽  
Arbitrary Dimensions ◽  
Restriction Inequality

Abstract Let $\mathbb {S}^{d-1}$ denote the unit sphere in Euclidean space $\mathbb {R}^d$ , $d\geq 2$ , equipped with surface measure $\sigma _{d-1}$ . An instance of our main result concerns the regularity of solutions of the convolution equation $$\begin{align*}a\cdot(f\sigma_{d-1})^{\ast {(q-1)}}\big\vert_{\mathbb{S}^{d-1}}=f,\text{ a.e. on }\mathbb{S}^{d-1}, \end{align*}$$ where $a\in C^\infty (\mathbb {S}^{d-1})$ , $q\geq 2(d+1)/(d-1)$ is an integer, and the only a priori assumption is $f\in L^2(\mathbb {S}^{d-1})$ . We prove that any such solution belongs to the class $C^\infty (\mathbb {S}^{d-1})$ . In particular, we show that all critical points associated with the sharp form of the corresponding adjoint Fourier restriction inequality on $\mathbb {S}^{d-1}$ are $C^\infty $ -smooth. This extends previous work of Christ and Shao [4] to arbitrary dimensions and general even exponents and plays a key role in the companion paper [24].

Fourier Restriction, Decoupling, and Applications

2019 ◽  
Author(s):  
Ciprian Demeter
Keyword(s):  
Fourier Restriction

scholarly journals Strichartz Estimates and Fourier Restriction Theorems on the Heisenberg Group

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Hajer Bahouri ◽  
Davide Barilari ◽  
Isabelle Gallagher
Keyword(s):  
Heisenberg Group ◽  
Strichartz Estimates ◽  
Fourier Restriction ◽  
Restriction Theorems

scholarly journals Improved estimates for the discrete Fourier restriction to the higher dimensional sphere

2013 ◽  
Vol 57 (1) ◽  
pp. 213-227 ◽  
Author(s):  
Jean Bourgain ◽  
Ciprian Demeter
Keyword(s):  
Dimensional Sphere ◽  
Fourier Restriction ◽  
Higher Dimensional

scholarly journals Time-Harmonic Solutions for Maxwell’s Equations in Anisotropic Media and Bochner–Riesz Estimates with Negative Index for Non-elliptic Surfaces

2021 ◽  
Author(s):  
Rainer Mandel ◽  
Robert Schippa
Keyword(s):  
Gaussian Curvature ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Smooth Surfaces ◽  
Negative Index ◽  
Elliptic Surfaces ◽  
Conical Singularities ◽  
Fourier Restriction ◽  
Time Harmonic ◽  
Homogeneous Media

AbstractWe solve time-harmonic Maxwell’s equations in anisotropic, spatially homogeneous media in intersections of $$L^p$$ L p -spaces. The material laws are time-independent. The analysis requires Fourier restriction–extension estimates for perturbations of Fresnel’s wave surface. This surface can be decomposed into finitely many components of the following three types: smooth surfaces with non-vanishing Gaussian curvature, smooth surfaces with Gaussian curvature vanishing along one-dimensional submanifolds but without flat points, and surfaces with conical singularities. Our estimates are based on new Bochner–Riesz estimates with negative index for non-elliptic surfaces.

scholarly journals Fourier restriction to polynomial curves I: a geometric inequality

2010 ◽  
Vol 132 (4) ◽  
pp. 1031-1076 ◽  
Author(s):  
Spyridon Dendrinos ◽  
James Wright
Keyword(s):  
Geometric Inequality ◽  
Polynomial Curves ◽  
Fourier Restriction

Problems of Harmonic Analysis Related to Finite-Type Hypersurfaces in R3, and Newton Polyhedra Detlef Müller

2014 ◽  
Keyword(s):  
Harmonic Analysis ◽  
Finite Type ◽  
Maximal Function ◽  
Oscillatory Integrals ◽  
The Other ◽  
Riemannian Surface ◽  
Fourier Restriction ◽  
Newton Polyhedra ◽  
The Given ◽  
Finite Type Hypersurfaces

This chapter presents three sets of problems and explains how these questions can be answered in an (almost) complete way in terms of Newton polyhedra associated to the given surface S (here, a smooth, finite type hypersurface in R³ with Riemannian surface measure dσ‎). The first problem is a classical question about estimates for oscillatory integrals, and there exists a huge body of results on it, in particular for convex hypersurfaces. The other two problems had first been formulated by Stein: the study of maximal averages along hypersurfaces has been initiated in Stein's work on the spherical maximal function, and also the idea of Fourier restriction goes back to him.

A Restriction Theorem for a k-Surface in ℝn

2005 ◽  
Vol 48 (2) ◽  
pp. 260-266 ◽  
Author(s):  
Daniel M. Oberlin
Keyword(s):  
Restriction Theorem ◽  
Fourier Restriction ◽  
Restriction Estimate

AbstractWe establish a sharp Fourier restriction estimate for a measure on a k-surface in ℝn, where n = k(k + 3)/2.

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