The multilinear restriction estimate: a short proof and a refinement

Multilinear Oscillatory Integrals Via Reduction of Dimension

Christ, Li, Tao, and Thiele have established multilinear oscillatory integral operator inequalities under severe dimensional restrictions. These restrictions are relaxed substantially in the present paper, but are by no means eliminated. The simple method used here has the advantage of yielding power type bounds, but cannot be expected to extend to the general case.

Restriction estimates to complex hypersurfaces

The restriction problem is better understood for hypersurfaces and recent progresses have been made by bilinear and multilinear approaches and most recently polynomial partitioning method which is combined with those estimates. However, for surfaces with codimension bigger than 1, bilinear and multilinear generalization of restriction estimates are more involved and effectiveness of these multilinear estimates is not so well understood yet. Regarding the restriction problem for the surfaces with codimensions bigger than 1, the current state of the art is still at the level of $TT^*$ method whi…

Bounds on oscillatory integral operators based on multilinear estimates

We apply the Bennett-Carbery-Tao multilinear restriction estimate in order to bound restriction operators and more general oscillatory integral operators. We get improved L^p estimates in the Stein restriction problem for dimension at least 5 and a small improvement in dimension 3. We prove similar estimates for Hormander-type oscillatory integral operators when the quadratic term in the phase function is positive definite, getting improvements in dimension at least 5. We also prove estimates for Hormander-type oscillatory integral operators in even dimensions. These last oscillatory estimates…

线性与多线性算子Stein定理的加权推广

Weighted Extensions of Stein's Theorem for Linear and Multilinear Operators

针对核函数仅满足尺寸条件的积分算子，在Herz和Cesàro型空间上建立加权估计，并推广到多线性强型和弱型情形，应用于粗糙奇异积分算子。

On multilinear determinant functionals

This paper considers the problem of $L^p$-estimates for a certain multilinear functional involving integration against a kernel with the structure of a determinant. Examples of such objects are ubiquitous in the study of Fourier restriction and geometric averaging operators. It is shown that, under very general circumstances, the boundedness of such functionals is equivalent to a geometric inequality for measures which has recently appeared in work by D. Oberlin (Math Proc. Cambridge. Philos. Soc., 129, 2000) and Bak, Oberlin, and Seeger (J. Aust. Math. Soc., 85, 2008).

Restriction Estimates for space curves with respect to general measures

In this paper we consider adjoint restriction estimates for space curves with respect to general measures and obtain optimal estimates when the curves satisfy a finite type condition. The argument here is new in that it doesn't rely on the \emph{offspring curve} method, which has been extensively used in the previous works. Our work was inspired by the recent argument due to Bourgain and Guth which was used to deduce linear restriction estimates from multilinear estimates for hypersurfaces.

$L^p$-加权锥限制估计

$L^p$-weighted cone restriction estimates

该研究针对截断锥的傅里叶延拓估计，在特定权重条件下，结合多线性精细Strichartz不等式等方法得到一系列界，并给出相关实例。

多卷积与多线性分形傅里叶扩展估计

Knapp-type obstructions in multilinear fractal Fourier extension

本文研究了多线性分形傅里叶扩展估计，提出了一种适用于卷积属于$ L^p $空间的测度的估计方法，并获得了Knapp型例子及两个必要条件。

任意余维二次流形的限制性估计

Restriction estimates for quadratic manifolds of arbitrary codimensions

本文将Bourgain-Guth多线性方法推广到任意余维二次流形，利用实代数几何中的覆盖引理与柱形分解算法，建立了统一的限制性估计并恢复了抛物面情形的主要定理。

Discrete Fourier restriction associated with Schrodinger equations

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 Schrödinger equations on torus. Some sharp estimates on $L^{\frac{2(d+2)}{d}}$ norm of certain exponential sums in higher dimensional cases are established. As an application, we show that some discrete multilinear maximal functions are bounded on $L^2(\mathbb Z)$.

Restriction of Fourier transforms to curves: An endpoint estimate with affine arclength measure

Consider the Fourier restriction operator associated to a curve in $R^d$, $d\ge 3$. We prove for various classes of curves the endpoint restricted strong type estimate with respect to affine arclength measure on the curve. An essential ingredient is an interpolation result for multilinear operators with symmetries acting on sequences of vector-valued functions.

Some remarks on Fourier restriction estimates

We provide $L^p \to L^q$ refinements on some Fourier restriction estimates obtained using polynomial partitioning. Let $S\subset \mathbb{R}^3$ be a compact $C^\infty$ surface with strictly positive second fundamental form. We derive sharp $L^p(S) \to L^q(\mathbb{R}^3)$ estimates for the associated Fourier extension operator for $q> 3.25$ and $q\geq 2p'$ from an estimate of Guth that was used to obtain $L^\infty(S) \to L^q(\mathbb{R}^3)$ bounds for $q>3.25$. We present a slightly weaker result when $S$ is the hyperbolic paraboloid in $\mathbb{R}^3$ based on the work of Cho and Lee. Finally, we …

Uniqueness in the Calderón problem and bilinear restriction estimates

Uniqueness in the Calderón problem in dimension bigger than two was usually studied under the assumption that conductivity has bounded gradient. For conductivities with unbounded gradients uniqueness results have not been known until recent years. The latest result due to Haberman basically relies on the optimal $L^2$ restriction estimate for hypersurface which is known as the Tomas-Stein restriction theorem. In the course of developments of the Fourier restriction problem bilinear and multilinear generalizations of the (adjoint) restriction estimates under suitable transversality condition be…

Restricted convolution inequalities, multilinear operators and applications

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 $$ {||(F*G)|_H||}_{L^{r}(H)} \leq {||F||}_{Λ^H_{2, p}({\Bbb R}^{n})} \cdot {||G||}_{Λ^H_{2, q}({\Bbb R}^{n})},$$ where the mixed norms on the right are defined by $$ {||F||}_{Λ^H_{2,p}({\Bbb R}^{n})}={(\int_{H^*} {(\int {|\hat{F}|}^2 dH_ξ^{\perp})}^{\frac{p}{2}} dξ)}^{\frac{1}{p}},$$ with $dH_ξ^{\perp}$ the $(n-k)$-dimensional Lebesgue measure on the affine subspace $H_ξ^{\perp}…

$\nmathbb{R}^4$ 中限制性与 Kakeya 问题范围的改进

Refinements on the Restriction and Kakeya problems range in $\mathbb{R}^4$

通过证明两个两端点关联估计，改进了 $\mathbb{R}^4$ 中 Fourier 限制性与 Kakeya 极大估计的范围，并探讨了对 Kakeya 集维数下界和 Bochner--Riesz 猜想的影响。

The proof of the $l^2$ Decoupling Conjecture

We prove the $l^2$ Decoupling Conjecture for compact hypersurfaces with positive definite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is the validity of the Discrete Restriction Conjecture, which (up to $N^ε$ losses) implies the full range of expected $L^p_{x,t}$ Strichartz estimates for both classical and irrational tori. Another one is an improvement in the range for the discrete restriction theory for lattice points on the sphere. Various applications in Additive Combinatorics, Incidence Geometry and Number Theory are also disc…

Incidence theory and restriction estimates

We investigate the interplay between the discrete restriction phenomenon and incidence theory. Two angles are explored. One is a refinement of the machinery developed by Thomas Wolff, which when combined with a recent subcritical estimate of Bourgain leads to new Strichartz estimates for irrational tori. The other one connects various additive energies with the Szemerédi-Trotter Theorem. The combination of the two approaches recovers the best known Strichartz estimates for the classical torus in dimensions $n\ge 3$, without any appeal to number theory.

A note on bilinear wave-Schrödinger interactions

We consider bilinear restriction estimates for wave-Schrödinger interactions and provided a sharp condition to ensure that the product belongs to $L^q_t L^r_x$ in the full bilinear range $\frac{2}{q} + \frac{d+1}{r} < d+1$, $1 \leqslant q, r \leqslant 2$. Moreover, we give a counter-example which shows that the bilinear restriction estimate can fail, even in the transverse setting. This failure is closely related to the lack of curvature of the cone. Finally we mention extensions of these estimates to adapted function spaces. In particular we give a general transference type principle for $U^2…