Related papers: A uniform Fourier restriction theorem for surfaces…

Using nonstandard methods, we show that the time dependent Fourier series of any smooth function F, solving the wave equation, on a finite closed interval, with vanishing boundary conditions, converges uniformly to F.

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.

We prove sharp endpoint results for the Fourier restriction operator associated to nondegenerate curves in $\Bbb R^d$, $d\ge 3$, and related estimates for oscillatory integral operators. Moreover, for some larger classes of curves in $\Bbb…

The problem of $L^p(R^3)\to L^2(S)$ Fourier restriction estimates for smooth hypersurfaces S of finite type in R^3 is by now very well understood for a large class of hypersurfaces, including all analytic ones. In this article, we take up…

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 improve the $L^p(\mathbb{R}^n)$ bounds on Stein's square function to the best-known range of the Fourier restriction problem when $n\geq4$. Applications including certain local smoothing estimates are also discussed.

The paper proves that a bound on the averaged Jones' square function of a measure implies an upper bound on the measure. Various types of assumptions on the measure are considered. The theorem is a generalization of a result due to A. Naber…

We prove a Wiener-type theorem for arcs in the unit circle which concerns express the measure of an arc in the unit circle via the measure's Fourier coefficients. Then we use it to give the Fourier series of the Cantor and to compute the…

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 $\mathbb{R}^{n+1}$.…

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…

Using the notion of higher-order Fourier dimension introduced in \cite{M2} (which was a sort of psuedorandomness condition stemming from the Gowers norms of Additive Combinatorics), we prove a maximal theorem and corresponding…

Consider the group ${\mathbb{R}}^2$ with the discrete topology, and denote its Fourier algebra by $A({{\mathbb{R}}_{\rm d}^2})$. We reformulate a theorem of V.A. Yudin as a statement about restrictions of functions in $A({{\mathbb{R}}_{\rm…

We prove a uniform estimate, valid for every closed Riemann surface of genus at least two, that bounds the distance of any quadratic differential to the finite dimensional space of holomorphic quadratic differentials in terms of its…

Let ${{\bf R}_{\mathbb{S}^{d-1}}}(p\to q)$ denote the best constant for the $L^p(\mathbb{R}^d)\to L^q(\mathbb{S}^{d-1})$ Fourier restriction inequality to the unit sphere $\mathbb{S}^{d-1}$, and let ${\bf R}_{\mathbb{S}^{d-1}} (p\to…

In this note, we prove the uniform resolvent estimate of the discrete Schr\"odinger operator with dimension three. To do this, we show a Fourier decay of the surface measure on the Fermi surface.

We prove Strichartz estimates for the semigroups associated to stratified and/or rotating inviscid geophysical fluids using Fourier restriction theory. We prove new results for rotating stratified fluids, and recover results from [16] for…

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…

A proof of the uniformization theorem of Riemann surface is given with only elementary properties of holomorphic functions and not using the paracompacity of the surface. This proof leans on an holomorphic version of the topological…

In this paper, we study the restriction estimate for a certain surface of finite type in $\mathbb{R}^3$, and partially improves the results of Buschenhenke-M\"{u}ller-Vargas. The key ingredients of the proof include the so called…

We give an effective uniform bound on the multigraded regularity of a subscheme of a smooth projective toric variety X with a given multigraded Hilbert polynomial. To establish this bound, we introduce a new combinatorial tool, called a…