On multilinear oscillatory integrals, nonsingular and singular

Bounds For Multilinear Sublevel Sets Via Szemeredi's Theorem

In 2005, Li, Tao, Thiele and the author raised a general question concerning upper bounds for a class of multilinear oscillatory integral operators, and established such bounds in a few cases. Most cases remain open. The present paper is concerned with sublevel set bounds, which would be a consequence of the oscillatory integral bounds, if valid. These sublevel set bounds are established here in a weak form but in nearly full generality, subject only to a rationality hypothesis. The proof relies on an extension of Szemeredi's theorem due to Furstenberg and Katznelson.

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。

Decay Rate of n-Linear Oscillatory Integral Operators in $\mathbb{R}^2$

In this paper, we prove $L^p$ decay estimates for multilinear oscillatory integrals in $\mathbb{R}^2$, establishing sharpness through a scaling argument. The result in this paper is a generalization of the previous work by Gressman and Xiao (2016).

A multilinear Fourier extension identity on $\mathbb{R}^n$

We prove an elementary multilinear identity for the Fourier extension operator on $\mathbb{R}^n$, generalising to higher dimensions the classical bilinear extension identity in the plane. In the particular case of the extension operator associated with the paraboloid, this provides a higher dimensional extension of a well-known identity of Ozawa and Tsutsumi for solutions to the free time-dependent Schrödinger equation. We conclude with a similar treatment of more general oscillatory integral operators whose phase functions collectively satisfy a natural multilinear transversality condition. T…

Maximal Decay Inequalities for Trilinear oscillatory integrals of convolution type

In this paper we prove sharp $L^\infty$-$L^\infty$-$L^\infty$ decay for certain trilinear oscillatory integral forms of convolution type on $\mathbb R^2$. These estimates imply earlier $L^2$-$L^2$-$L^2$ results obtained by the second author as well as corresponding sharp, stable sublevel set estimates of the form studied by Christ and Christ, Li, Tao, and Thiele. New connections to the multilinear results of Phong, Stein, and Sturm are also considered.

Sharp estimates for trilinear oscillatory integrals and an algorithm of two-dimensional resolution of singularities

We obtain sharp estimates for certain trilinear oscillatory integrals. In particular, we extend Phong and Stein's seminal result to a trilinear setting. This result partially answers a question raised by Christ, Li, Tao and Thiele concerning the sharp estimates for certain multilinear oscillatory integrals. The method in this paper relies on a self-contained algorithm of resolution of singularities in $\mathbb R^2$, which may be of independent interest.

Oscillatory Loomis-Whitney and Projections of Sublevel Sets

We consider an oscillatory integral operator with Loomis-Whitney multilinear form. The phase is real analytic in a neighborhood of the origin in $\mathbb{R}^d$ and satisfies a nondegeneracy condition related to its Newton polyhedron. Maximal decay is obtained for this operator in certain cases, depending on the Newton polyhedron of the phase and the given Lebesgue exponents. Our estimates imply volumes of sublevel sets of such real analytic functions are small relative to the product of areas of projections onto coordinate hyperplanes.

振荡积分与谱簇的尖锐端点多线性估计

Sharp endpoint multilinear estimates for oscillatory integrals and spectral clusters

本文证明了任意分离频率尺度下Carleson–Sjölin振荡积分算子的尖锐k线性L^p估计，并解决了三维闭Riemann流形上Burq–Gérard–Tzvetkov的对数自由端点双线性谱簇估计问题。

多线性算子的多尺度分析

Multi-scale analysis for multilinear operators

该研究针对多线性Calderón–Zygmund理论外的多线性算子，构建统一多尺度框架，建立稀疏形式界并推导加权范数不等式，应用于多线性粗糙奇异积分等算子的稀疏控制与界的研究。

Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces

Using the multilinear theory of oscillatory integral operators, we establish a square function inequality with implications to moment inequalities for discrete exponential sums which frequencies lie on curved hyper-surfaces. In particular a new distributional estimate for toral eigenfunctions is established.

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}…