Related papers: Some inequalities for the Fourier transform and th…

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 new Pitt inequalities for the Fourier transforms with radial and non-radial weights using weighted restriction inequalities for the Fourier transform on the sphere. We also prove new Riemann-Lebesgue estimates and versions of the…

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…

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…

We deal with the restriction phenomenon for the Fourier transform. We prove that each of the restriction conjectures for the sphere, the paraboloid, the elliptic hyperboloid in $\mathbb{R}^n$ implies that for the cone in $\mathbb{R}^{n+1}$.…

In this paper, we present a refined version of the (classical) Stein inequality for the Fourier transform, elevating it to a new level of accuracy. Furthermore, we establish extended analogues of a more precise version of the Stein…

Limiting real interpolation method is applied to describe the behaviour of the Fourier coefficients of functions that belong to spaces which are "very close" to L2.

We use a restriction theorem for Fourier transforms of fractal measures to study projections onto families of planes in R^3 whose normal directions form nondegenerate curves.

We prove new Fourier restriction estimates to the unit sphere $S^{d-1}$ on the class of $O(d-k)\times O(k)$-symmetric functions, for every $d\geq 4$ and $2\leq k\leq d-2$. As an application, we establish the existence of maximizers for the…

This article explores weighted $(L^p, L^q)$ inequalities for the Fourier transform in rank one Riemannian symmetric spaces of noncompact type. We establish both necessary and sufficient conditions for these inequalities to hold. To prove…

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.

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.…

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

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.

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…

The adjoint Fourier restriction inequality of Tomas and Stein states that the mapping $f\mapsto \widehat{f\sigma}$ is bounded from $\lt(S^2)$ to $L^4(\reals^3)$. We prove that there exist functions which extremize this inequality, and that…

In this article, we study the problem of obtaining Lebesgue space inequalities for the Fourier restriction operator associated to rectangular pieces of the paraboloid and perturbations thereof. We state a conjecture for the dependence of…

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…

In this article, we define the Fourier-Dunkl transform, which generalizes the Fourier transform. We prove Strichartz's restriction theorem for the Fourier-Dunkl transform for a cone-hyper-surface and its generalisation to the family of…

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…