Related papers: $L^2$ restriction estimates from the Fourier spect…

This is the first of two articles in which we prove a sharp $L^p-L^2$ Fourier restriction theorem for a large class of smooth, finite type hypersurfaces in $\Bbb R^3$, which includes in particular all real-analytic hypersurfaces. The…

The purpose of this paper is to prove a Fourier restriction estimate for certain 2-dimensional surfaces in $\bbR^{2d}$, $d\ge 3$. These surfaces are defined by a complex curve $\gamma(z)$ of simple type, which is given by a mapping of the…

The Fourier spectrum is a family of dimensions that interpolates between the Fourier and Hausdorff dimensions and are defined in terms of certain energies which capture Fourier decay. In this paper we obtain a convenient discrete…

We prove uniform $L^p \to L^q$ bounds for Fourier restriction to polynomial curves in $\mathbb R^d$ with affine arclength measure, in the conjectured range.

We explore the extent to which the Fourier transform of an $L^p$ density supported on the sphere in $\mathbb{R}^n$ can have large mass on affine subspaces, placing particular emphasis on lines and hyperplanes. This involves establishing…

Recently Wolff obtained a nearly sharp $L^2$ bilinear restriction theorem for bounded subsets of the cone in general dimension. We obtain the endpoint of Wolff's estimate and generalize to the case when one of the subsets is large. As a…

In this paper we prove a sharp trilinear inequality which is motivated by a program to obtain the sharp form of the $L^2 - L^6$ Tomas-Stein adjoint restriction inequality on the circle. Our method uses intricate estimates for integrals of…

We give a new proof of a classic Fourier restriction theorem for the truncated paraboloid in $\mathbb{R}^n$ based on the $l^2$ decoupling theorem of Bourgain-Demeter. Focusing on the extension formulation of the restriction problem (dual to…

We study the restriction of the Fourier transform to quadratic surfaces in vector spaces over finite fields. In two dimensions, we obtain the sharp result by considering the sums of arbitrary two elements in the subset of quadratic surfaces…

In this article, we prove the restriction theorem for the Fourier-Hermite transform and obtain the Strichartz estimate for the system of orthonormal functions for the Hermite operator $H=-\Delta+|x|^2$ on $\mathbb{R}^n$ as application.…

Let $S \subset \Bbb R^n$ be a smooth compact hypersurface with a strictly positive second fundamental form, $E$ be the Fourier extension operator on $S$, and $X$ be a Lebesgue measurable subset of $\Bbb R^n$. If $X$ contains a ball of each…

We prove a limit theorem for quantum stochastic differential equations with unbounded coefficients which extends the Trotter-Kato theorem for contraction semigroups. From this theorem, general results on the convergence of approximations…

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…

We prove a Central Limit Theorem for the linear statistics of two-dimensional Coulomb gases, with arbitrary inverse temperature and general confining potential, at the macroscopic and mesoscopic scales and possibly near the boundary of the…

In Fourier restriction problems, a cone and a paraboloid are model surfaces. The sharp bilinear cone restriction estimate was first shown by Wolff, and later the endpoint was obtained by Tao. For a paraboloid, the sharp $L^2$ bilinear…

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…

We prove $L^p \rightarrow L^q$ Fourier restriction estimates for 3-dimensional quadratic surfaces in $\mathbb{R}^5$. Our results are sharp, up to endpoints, for a few classes of surfaces.

In a recent paper [Ann. of Math. 189 (2019), 837--861], Du and Zhang proved a fractal Fourier restriction estimate and used it to establish the sharp $L^2$ estimate on the Schr\"{o}dinger maximal function in $\Bbb R^n$, $n \geq 2$. In this…

We systematically study weighted $L^2$ restriction for quadratic manifolds of arbitrary codimensions by sharp uniform Fourier decay estimates and a refinement of the Du-Zhang method. Comparison with prior results is also discussed. In…

We prove an $L^2\times L^2\to L^q_tL^r_x$ bilinear adjoint Fourier restriction estimate for $n$-dimensional elliptic paraboloids, with $n\ge 2$ and $1\le q \le \infty$, $1\le r\le 2$ being on the endline…