Terence Tao
Recently Wolff obtained a nearly sharp $L^2$ bilinear restriction theorem for bounded subsets of the cone in general dimension. We obtain the endpoint of Wolff's estimate and generalize to the case when one of the subsets is large. As a consequence, we are able to deduce some nearly-sharp $L^p$ null form estimates.
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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 restriction estimate was obtained by Tao, but the endpoint was remained open. In this paper we prove the endpoint $L^2$ bilinear restriction estimate for a paraboloid.
The endpoint Strichartz estimate $\| e^{itΔ} f \|_{L^2_t L^\infty_x(\R \times \R^2)} \lesssim \|f\|_{L^2_x(\R^2)}$ is known to be false by the work of Montgomery-Smith, despite being only ``logarithmically far'' from being true in some sense. In this short note we show that (in sharp constrast to the $L^p_{t,x}$ Strichartz estimates) the situation is not improved by passing to a bilinear setting; more precisely, if $P, P'$ are non-trivial smooth Fourier cutoff multipliers then we show that the bilinear estimate $$\| (e^{itΔ} P f) (e^{itΔ} P' g) \|_{L^2_t L^\infty_x(\R \times \R^2)} \lesssim \|…
This note introduces bilinear estimates intended as a step towards an $L^\infty$-endpoint Kato-Ponce inequality. In particular, a bilinear version of the classical Gagliardo-Nirenberg interpolation inequalities for a product of functions is proved.
In this paper, we establish some bilinear endpoint estimates of Calderón commutator $\mathcal{C}[\nabla A,f](x)$ with a homogeneous kernel when $Ω\in L\log^+L(\mathbf{S}^{d-1})$. More precisely, we prove that $\mathcal{C}[\nabla A,f]$ maps $L^q(\mathbb{R}^d)\times L^1(\mathbb{R}^d)$ to $L^{r,\infty}(\mathbb{R}^d)$ if $q>d$ which improves previous result essentially. If $q=d$, we show that Calderón commutator maps $L^{d,1}(\mathbb{R}^d)\times L^1(\mathbb{R}^d)$ to $L^{r,\infty}(\mathbb{R}^d)$ which is new even if the kernel is smooth. The novelty in the paper is that we prove a new endpoint est…
Endpoint Criteria for One-Dimensional Bilinear Rough Singular Integrals
研究一维双线性粗糙奇异积分的端点定理，通过刻画相关角乘子结构，在临界核假设下建立两个有界性准则，包括\(L\log L\)结果及临界方向指标下的结果，且两类临界核不可比，方法涉及多种数学工具。
We establish a sharp bilinear estimate for the Klein--Gordon propagator in the spirit of recent work of Beltran--Vega. Our approach is inspired by work in the setting of the wave equation due to Bez, Jeavons and Ozawa. As a consequence of our main bilinear estimate, we deduce several sharp estimates of null-form type and recover some sharp Strichartz estimates found by Quilodrán and Jeavons.
We prove (adjoint) bilinear restriction estimates for general phases at different scales in the full non-endpoint mixed norm range, and give bounds with a sharp and explicit dependence on the phases. These estimates have applications to high-low frequency interactions for solutions to partial differential equations, as well as to the linear restriction problem for surfaces with degenerate curvature. As a consequence, we obtain new bilinear restriction estimates for elliptic phases and wave/Klein-Gordon interactions in the full bilinear range, and give a refined Strichartz inequality for the Kl…
We study the behavior of the bilinear Hilbert transform $\mathrm{BHT}$ at the boundary of the known boundedness region $\mathcal H$. A sample of our results is the estimate $| \langle\mathrm{BHT}(f_1,f_2),f_3 \rangle | \leq C |F_1|^{\frac34}|F_2| ^{\frac34} |F_3|^{-\frac12} \log\log \Big(\mathrm{e}^{\mathrm{e}} + \frac{|F_3|}{\min\{|F_1|,|F_2|\}} \Big) $ valid for all tuples of sets $F_j \subset \mathbb R $ of finite measure and functions $f_j$ such that $|f_j| \leq \mathbf{1}_{F_j}$, $j=1,2,3$, with the additional restriction that $f_3$ be supported on a major subset $F_3'$ of $F_3$ that depe…
Sharp endpoint $L^p-$estimates for Bilinear spherical maximal functions
本文研究双线性球面极大函数的端点估计，在低维情形得到临界限制弱型估计，并由此推出多线性球面极大函数的尖锐端点估计，同时改进局部球面极大函数的 L^p 有界性并给出线性化版本的尖锐结果。
Strong-type endpoint obstructions for bilinear spherical maximal operators
该研究针对维度 $d\geq2$ 的双线性球面极大算子，证明其间隙极大算子在特定指数条件下不满足有界性，给出洛伦兹指数准则并推广负结果至多线性球面极大算子，明确适用的强型边界范围。
Bilinear oscillatory Fourier multipliers
本文研究含振荡因子的双线性 Fourier 乘子，建立其在 Lebesgue 空间间（含端点）的有界性，并讨论所得结果的尖锐性。
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. Except for an endpoint, this answers a conjecture of Machedon and Klainerman, and also improves upon the known restriction theory for the paraboloid and sphere.
Input-Endpoint Compactness and Essential Norms for Weighted Bilinear Hardy Operators
本文研究加权双线性 Hardy 算子在输入端点（指数含1或无穷）的紧性，给出临界指数范围、活动尾部衰减刻画及本质范数公式，并处理拟 Banach 目标与有限端点情形。
We prove certain endpoint restriction estimates for the paraboloid over finite fields in three and higher dimensions. Working in the bilinear setting, we are able to pass from estimates for characteristic functions to estimates for general functions while avoiding the extra logarithmic power of the field size which is introduced by the dyadic pigeonhole approach. This allows us to remove logarithmic factors from the estimates obtained by Mockenhaupt and Tao in three dimensions and those obtained by Iosevich and Koh in higher dimensions.
The purpose of this article is to provide an alternative proof of the weak-type $\left(1,\ldots,1;\frac{1}{m}\right)$ estimate for $m$-multilinear Calderón-Zygmund operators on $\mathbb{R}^n$ first proved by Grafakos and Torres. Subsequent proofs in the bilinear setting have been given by Maldonado and Naibo and also by Pérez and Torres. The proof given here is motivated by the proof of the weak-type $(1,1)$ estimate for Calderón-Zygmund operators in the nonhomogeneous setting by Nazarov, Treil, and Volberg.
A bilinear approach to the finite field restriction problem, II
本文通过双线性方法与点线关联估计，证明三维抛物面P3的傅里叶扩展算子从L^2到L^r（r>176/51）的有界性，并得到六维抛物面P6的L^2到L^{8/3}的端点估计。
It is a result by Lacey and Thiele that the bilinear Hilbert transform maps L^{p_1}(R) \times L^{p_2}(R) into L^{p_3}(R) whenever (p_1,p_2,p_3) is a Holder tuple with p_1,p_2 > 1 and p_3>2/3. We study the behavior of the quartile operator, which is the Walsh model for the bilinear Hilbert transform, when p_3=2/3. We show that the quartile operator maps L^{p_1}(R) \times L^{p_2}(R) into L^{2/3,\infty}(R) when p_1,p_2>1 and one component is restricted to subindicator functions. As a corollary, we derive that the quartile operator maps L^{p_1}(R) \times L^{p_2,2/3}(R) into L^{2/3,\infty}(R). We a…
Bilinear Rough Singular Integrals near the Critical Integrability via Sharp Fourier Multiplier Criteria
本文研究了双线性粗糙奇异积分算子在临界可积性附近的有界性问题，提出了一种新的傅里叶乘子算子有界性准则，改进了已有条件并接近线性理论推测的端点估计。
Compactness and Essential Norms of Weighted Bilinear Hardy Operators on General Intervals
本文研究一般区间上加权双线性 Hardy 算子的紧性，证明其紧性等价于有界性与端点尾部拟范数消失，并给出本质范数、有限秩距离与非紧性测度的精确恒等式，以及显式紧性判据和逼近数收敛结果。
The aim of this article is to give a complete solution to the problem of the bilinear decompositions of the products of some Hardy spaces $H^p(\mathbb{R}^n)$ and their duals in the case when $p<1$ and near to $1$, via wavelets, paraproducts and the theory of bilinear Calderón-Zygmund operators. Precisely, the authors establish the bilinear decompositions of the product spaces $H^p(\mathbb{R}^n)\times\dotΛ_α (\mathbb{R}^n)$ and $H^p(\mathbb{R}^n)\timesΛ_α(\mathbb{R}^n)$, where, for all $p\in(\frac{n}{n+1},\,1)$ and $α:=n(\frac{1}{p}-1)$, $H^p(\mathbb{R}^n)$ denotes the classical real Hardy spac…
A characterization of compactness via bilinear $T1$ theorem
本文通过双线性$T1$定理刻画了双线性Calderón-Zygmund算子的加权紧性，证明了其紧性等价于核紧性、弱紧性及CMO条件，并基于二进表示与外推方法给出了端点紧性准则。
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。
We prove that in dimensions $d \geq 3$, the non-endpoint, Lorentz-invariant $L^2 \to L^p$ adjoint Fourier restriction inequality on the $d$-dimensional hyperboloid $\mathbb{H}^d \subseteq \mathbb{R}^{d+1}$ possesses maximizers. The analogous result had been previously established in dimensions $d=1,2$ using the convolution structure of the inequality at the lower endpoint (an even integer); we obtain the generalization by using tools from bilinear restriction theory.
Endpoint estimates for the fractal circular maximal function and related local smoothing
本文证明了分形圆极大函数在伸缩集分形维数位于 $[1/2, 1)$ 时缺失的端点估计，并利用双线性方法扩展了分形伸缩集上波动算子局部光滑性估计成立的 $p,q$ 范围。
Let $L= -Δ+ V$ be a Schrödinger operator on $\mathbb R^d$, $d\geq 3$, where $V$ is a nonnegative potential, $V\ne 0$, and belongs to the reverse Hölder class $RH_{d/2}$. In this paper, we study the commutators $[b,T]$ for $T$ in a class $\mathcal K_L$ of sublinear operators containing the fundamental operators in harmonic analysis related to $L$. More precisely, when $T\in \mathcal K_L$, we prove that there exists a bounded subbilinear operator $\mathfrak R= \mathfrak R_T: H^1_L(\mathbb R^d)\times BMO(\mathbb R^d)\to L^1(\mathbb R^d)$ such that $|T(\mathfrak S(f,b))|- \mathfrak R(f,b)\leq |[b,…
Let $Ω$ be homogeneous of degree zero, have vanishing moment of order one on the unit sphere $\mathbb {S}^{d-1}$($d\ge 2$). In this paper, our object of investigation is the following rough non-standard singular integral operator $$T_{Ω,\,A}f(x)={\rm p.\,v.}\int_{\mathbb{R}^d}\frac{Ω(x-y)}{|x-y|^{d+1}}\big(A(x)-A(y)-\nabla A(y)(x-y)\big)f(y){\rm d}y,$$ where $A$ is a function defined on $\mathbb{R}^d$ with derivatives of order one in ${\rm BMO}(\mathbb{R}^d)$. We show that $T_{Ω,\,A}$ enjoys the endpoint $L\log L$ type estimate and is $L^p$ bounded if $Ω\in L(\log L)^{2}(\mathbb{S}^{d-1})$. Th…
Variational Estimates for Bilinear Ergodic Averages Along Sublinear Sequences
本文证明了沿次线性序列的双线性遍历平均在任意保测系统上的长变分估计，覆盖完整预期范围并在端点处达到最佳，并由此导出双线性极大估计。
We prove that bilinear forms associated to the rough homogeneous singular integrals $T_Ω$ on $\mathbb R^d$, where the angular part $Ω\in L^q (S^{d-1})$ has vanishing average and $1<q\leq \infty$, and to Bochner-Riesz means at the critical index in $\mathbb R^d$ are dominated by sparse forms involving $(1,p)$ averages. This domination is stronger than the weak-$L^1$ estimates for $T_Ω$ and for Bochner-Riesz means, respectively due to Seeger and Christ. Furthermore, our domination theorems entail as a corollary new sharp quantitative $A_p$-weighted estimates for Bochner-Riesz means and for homog…
We prove that the weak-$L^{p}$ norms, and in fact the sparse $(p,1)$-norms, of the Carleson maximal partial Fourier sum operator are $\lesssim (p-1)^{-1}$ as $p\to 1^+$. This is an improvement on the Carleson-Hunt theorem, where the same upper bound on the growth order is obtained for the restricted weak-$L^p$ type norm, and which was the strongest quantitative bound prior to our result. Furthermore, our sparse $(p,1)$-norms bound imply new and stronger results at the endpoint $p=1$. In particular, we obtain that the Fourier series of functions from the weighted Arias de Reyna space $ QA_{\inf…
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