Related papers: Restriction estimates for the flat disks over fini…

We study two-dimensional spatio-spectral limiting operators \[ T_R := P_{D(R)} B_S P_{D(R)} : L^2(\mathbb{R}^2) \rightarrow L^2(\mathbb{R}^2), \] where $D(R)$ is a disk of radius $R>1$, $S\subset\mathbb{R}^2$ is a domain with well-shaped…

We show that one can obtain improved $L^4$ geodesic restriction estimates for eigenfunctions on compact Riemannian surfaces with nonpositive curvature. We achieve this by adapting Sogge's strategy in proving improved critical $L^p$…

We obtain some sharp $L^p$ weighted Fourier restriction estimates of the form $\|Ef\|_{L^p(B^{n+1}(0,R),Hdx)} \lessapprox R^{\beta}\|f\|_2$, where $E$ is the Fourier extension operator over the truncated paraboloid, and $H$ is a weight…

It is well-known that the Fourier extension operator for the paraboloid in $\mathbb{R}^d$ cannot be weak-type bounded at the restriction endpoint $q = 2d/(d-1)$, since such an estimate would imply bounds for the Kakeya maximal function…

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 weighted versions of the 2D Restriction Conjecture for the unit sphere in $\mathbb{R}^2$. Our results involve the weight functions $(1+|x|)^\alpha(1+|y|)^\beta$ and $(1+|x|+|y|)^\gamma$ with $\alpha,\beta,\gamma\geq 0$.

We show that Blokhuis' quadratic upper bound for two-distance sets is sharp over finite fields in almost all dimensions. Our construction complements Lison\v{e}k's higher-dimensional maximal constructions that were carried out in Lorentz…

In this paper we find the sharp forms and characterize the complex-valued extremizers of the adjoint Fourier restriction inequalities on the sphere $$\big\|\widehat{f \sigma}\big\|_{L^{p}(\mathbb{R}^{d})} \lesssim…

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…

We derive lower bounds in best rational approximation of given degree to finite Blaschke products, in the Hardy space $H^2$ of the unit disk. We first consider approximation to $z^N$, and then move on to more general Blaschke products whose…

The purpose of these notes is describe the state of progress on the restriction problem in harmonic analysis, with an emphasis on the developments of the past decade or so on the Euclidean space version of these problems for spheres and…

Fourier restriction theorems, whose study had been initiated by E.M. Stein, usually describe a family of a priori estimates of the L^q-norm of the restriction of the Fourier transform of a function f in L^p (say, on Euclidean space) to a…

Let $\{u_\lambda\}$ be a sequence of $L^2$-normalized Laplacian eigenfunctions on a compact two-dimensional smooth Riemanniann manifold $(M,g)$. We seek to get an $L^p$ restriction bounds of the Neumann data $ \lambda^{-1} \partial_\nu…

We consider Fourier transforms of densities supported on curves in R^d. We obtain sharp lower and close to sharp upper bounds for the L^q decay rates.

We introduce a continuous analog of the Fourier ratio for compactly supported Borel measures. For a measure \(\mu\) on \(\mathbb{R}^d\) and \(f\in L^2(\mu)\), the Fourier ratio compares \(L^1\) and \(L^2\) norms of a regularized Fourier…

We improve the exponent for the discrete Fourier restriction to the $n$ dimensional sphere, from $p=\frac{2(n+1)}{n-3}$ to $p=\frac{2n}{n-3}$, when $n\ge 4$.

We prove new $L^p$-$L^q$-estimates for solutions to elliptic differential operators with constant coefficients in $\mathbb{R}^3$. We use the estimates for the decay of the Fourier transform of particular surfaces in $\mathbb{R}^3$ with…

Volumes of moduli spaces of hyperbolic cone surfaces were previously defined and computed when the angles of the cone singularities are at most 2pi. We propose a general definition of these volumes without restriction on the angles. This…

The first purpose of this paper is to solve completely the finite field cone restriction conjecture in four dimensions with $-1$ non-square. The second is to introduce a new approach to study incidence problems via restriction theory. More…

Let \psi be a Hecke-Maass form on a cubic division algebra over \Q. We apply arithmetic amplification to improve the local bound for the L^2 norm of \psi restricted to maximal flat subspaces.