Related papers: Stein-Tomas Restriction Theorem via Spectral Measu…

The classical Stein-Tomas restriction theorem is equivalent to the statement that the spectral measure $dE(\lambda)$ of the square root of the Laplacian on $\RR^n$ is bounded from $L^p(\RR^n)$ to $L^{p'}(\RR^n)$ for $1 \leq p \leq…

The Stein--Tomas restriction theorem is an important result in Fourier restriction theory. It gives a range of $q$ for which $L^q\to L^2$ restriction estimates hold for a given measure, in terms of the Fourier and Frostman dimensions of the…

We prove a maximal Fourier restriction theorem for the sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^{d}$ for any dimension $d\geq 3$ in a restricted range of exponents given by the Stein-Tomas theorem. The proof consists of a simple…

Mockenhaupt and Tao (Duke 2004) proved a finite field analogue of the Stein--Tomas restriction theorem, establishing a range of $q$ for which $L^q\to L^2$ restriction estimates hold for a given measure $\mu$ on a vector space over a finite…

We prove a Fourier restriction result, uniform over a certain collection of reference measures, for some indices in the Stein-Tomas range.

We present an elementary approach to prove restriction theorems for particular surfaces for which the Tomas-Stein theorem does not apply, which in turn provide short proofs for well-known Strichartz estimates for associated PDEs. The method…

In this paper, we prove a spectral restriction theorem on the three-dimensional Heisenberg nilmanifold. Since this manifold is an $\mathbb S^1$-bundle over the flat torus $\mathbb T^2$, the result provides a sub-elliptic counterpart of…

We prove an endpoint version of the Stein-Tomas restriction theorem, for a general class of measures, and with a strengthened Lorentz space estimate. A similar improvement is obtained for Stein's estimate on oscillatory integrals of…

We identify a one-parameter family of inequalities for the Fourier transform whose limiting case is the restriction conjecture for the sphere. Using Stein's method of complex interpolation we prove the conjectured inequalities when the…

We study L^p-L^r restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension $d$ is even, then it is conjectured that the L^{(2d+2)/(d+3)}-L^2 Stein-Tomas…

In this article we prove the existence of sets $E \subseteq \mathbb{R}$ of zero Fourier dimension such that it is possible to restrict the Fourier transform to $E$ on a certain non-trivial range $[1,\tilde{p})$ with $1<\tilde{p}<2$. This…

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.

We study spectral synthesis for measures supported on thin subsets of compact Riemannian manifolds. We prove that under natural non-concentration conditions, such measures admit quantitative spectral synthesis, with explicit stability…

We study $L^2$ to $L^p$ operator norms of spectral projectors for the Euclidean Laplacian on the torus in the case where the spectral window is narrow. With a window of constant size this is a classical result of Sogge; in the small-window…

This dissertation studies the Fourier restriction, which is to find the range of the constants p, q such that the L^q norm on a chosen subset of the Fourier domain is bounded above by the L^p norm in a spacial domain, up to some constant…

The Tomas-Stein inequality for a compact subset $\Gamma$ of the sphere $S^d$ states that the mapping $f\mapsto \widehat{f\sigma}$ is bounded from $L^2(\Gamma,\sigma)$ to $L^{2+4/d}(\R^{d+1})$. Then conditional on a strict comparison between…

The classical Stein--Tomas theorem extends the theory of linear Fourier restriction estimates from smooth manifolds to fractal measures exhibiting Fourier decay. In the multilinear setting, transversality allows for Fourier extension…

We establish uniformization results for metric spaces that are homeomorphic to the euclidean plane or sphere and have locally finite Hausdorff 2-measure. Applying the geometric definition of quasiconformality, we give a necessary and…

The Fourier restriction problem asks when it is meaningful to restrict the Fourier transform of a function to a given set. Many of the key examples are smooth co-dimension 1 manifolds, although there is increasing interest in fractal sets.…

In this note we discuss the Stein restriction problem on arbitrary $n$-torus, $n\geq 2$. In contrast with the usual cases of the sphere, the parabola and the cone, we provide necessary and sufficient conditions on the Lebesgue indices, by…