Related papers: Sharpness of the Mockenhaupt-Mitsis-Bak-Seeger Res…

We prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem in all dimensions $d$ and the full parameter range $0 < a,b < d$. Our construction is deterministic and also yields Salem sets.

We prove that the range of exponents in Mockenhaupt's restriction theorem for Salem sets, with the endpoint estimate due to Bak and Seeger, is optimal.

We use a deterministic construction to prove the optimality of the exponent in the Mockenhaupt-Mitsis-Bak-Seeger Fourier restriction theorem for dimension $d=1$ and parameter range $0 < a,b \leq d$ and $b\leq 2a$. Previous constructions by…

We construct Salem sets on the real line with endpoint Fourier decay and near-endpoint regularity properties. This complements a result of \L aba and Pramanik, who obtained near-endpoint Fourier decay and endpoint regularity properties. We…

We study restriction problem in vector spaces over finite fields. We obtain finite field analogue of Mockenhaupt-Mitsis-Bak-Seenger restriction theorem, and we show that the range of the exponentials is sharp.

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 sharp $L^2$ Fourier restriction inequalities for compact, smooth surfaces in $\mathbb{R}^3$ equipped with the affine surface measure or a power thereof. The results are valid for all smooth surfaces and the bounds are uniform for…

This is the second 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 R^3, which includes in particular all real-analytic hypersurfaces.

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 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…

An $L^2$ Fourier restriction argument of Bak and Seeger is abstracted to the setting of locally compact abelian groups. This is used to prove new restriction estimates for varieties lying in modules over local fields or rings of integers…

A Fourier restriction estimate is obtained for a broad class of conic surfaces by adding a weight to the usual underlying measure. The new restriction estimate exhibits a certain affine-invariance and implies the sharp $L^p-L^q$ restriction…

In this paper we study extension theorems associated with general varieties in two dimensional vector spaces over finite fields. Applying Bezout's theorem, we obtain the sufficient and necessary conditions on general curves where sharp…

Recently Wolff obtained a sharp $L^2$ bilinear restriction theorem for bounded subsets of the cone in general dimension. Here we adapt the argument of Wolff to also handle subsets of ``elliptic surfaces'' such as paraboloids and spheres.…

In this paper, we determine the sharp \((p,q)\) range for \(L^p\)--\(L^q\) bounds of convolution operators \(f\mapsto \mu*f\) associated with fractal measures \(\mu\in \mathcal P_{\alpha,\beta}(\mathbb R^d)\), namely, compactly supported…

We show that, for $n\geq 3$, $\lim_{t \to 0} e^{it\Delta}f(x) = f(x)$ holds almost everywhere for all $f \in H^s (\mathbb{R}^n)$ provided that $s>\frac{n}{2(n+1)}$. Due to a counterexample by Bourgain, up to the endpoint, this result is…

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.

In this paper, we present a different proof on the discrete Fourier restriction. The proof recovers Bourgain's level set result on Strichartz estimates associated with Schr\"odinger equations on torus. Some sharp estimates on…

The Fourier restriction conjecture is a fundamental problem in harmonic analysis. In this paper, we investigate restriction estimates for degenerate higher codimensional quadratic surfaces and obtain sharp results for some types of…

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…