Related papers: Fourier Restriction: From Linear Restriction to Mu…

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…

Using a bilinear restriction theorem of Lee and a bilinear-to-linear argument of Stovall, we obtain the conjectured range of Fourier restriction estimates for a conical hypersurface in $\mathbb{R}^4$ with hyperbolic cross sections.

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 provide a general scheme for proving $L^p$ estimates for certain bilinear Fourier restrictions outside the locally $L^2$ setting. As an application, we show how such estimates follow for the lacunary polygon. In contrast with prior…

Bilinear restriction estimates have been appeared in work of Bourgain, Klainerman, and Machedon. In this paper we develop the theory of these estimates (together with the analogues for Kakeya estimates). As a consequence we improve the…

In connection with the restriction problem in $\mathbb R^n$ for hypersurfaces including the sphere and paraboloid, the bilinear (adjoint) restriction estimates have been extensively studied. However, not much is known about such estimates…

Conditional on Fourier restriction estimates for elliptic hypersurfaces, we prove optimal restriction estimates for polynomial hypersurfaces of revolution for which the defining polynomial has non-negative coefficients. In particular, we…

We prove an $L^2 \times L^2 \rightarrow L_t^qL_x^p $ bilinear Fourier extension estimate for the cone when $p,q$ are on the critical line $1/q=(\frac{n+1}{2})(1-1/p)$. This extends previous results by Wolff, Tao and Lee-Vargas.

In contrast to elliptic surfaces, the Fourier restriction problem for hypersurfaces of non-vanishing Gaussian curvature which admit principal curvatures of opposite signs is still hardly understood. In fact, even for 2-surfaces, the only…

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…

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…

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…

We identify a one-parameter family of inequalities for the Fourier transform whose limiting case is the restriction conjecture for the sphere. Using Stein's method of complex interpolation we prove the conjectured inequalities when the…

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.

For $ 1\le k <n$, we prove that for functions $F,G$ on $ {\Bbb R}^{n}$, any $k$-dimensional affine subspace $H \subset {\Bbb R}^{n}$, and $p,q,r \ge 2$ with $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1$, one has the estimate $$…

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…

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.…

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 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…

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.