Giacomo Del Nin, Andrea Merlo
We show that if a measure of dimension $s$ on $\mathbb{R}^d$ admits $(p,q)$ Fourier restriction for some endpoint exponents allowed by its dimension, namely $q=\tfrac{s}{d}p'$ for some $p>1$, then it is either absolutely continuous or $1$-purely unrectifiable.
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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.
We revisit the multilinear Kakeya, curved Kakeya, restriction, and oscillatory integral estimates that were obtained in paper of Bennett, Carbery, and the author using a heat flow monotonicity method applied to a fractional Cartesian product, together with induction on scales arguments. Many of these estimates contained losses of the form $R^\varepsilon$ (or $\log^{O(1)} R$) for some scale factor $R$. By further developing the heat flow method, and applying it directly for the first time to the multilinear curved Kakeya and restriction settings, we are able to eliminate these losses, as long a…
We construct Salem sets on the real line with endpoint Fourier decay and near-endpoint regularity properties. This complements a result of Łaba and Pramanik, who obtained near-endpoint Fourier decay and endpoint regularity properties. We then modify the construction to extend a theorem of Hambrook and Łaba to show sharpness of the $L^2$-Fourier restriction estimate by Mockenhaupt and Bak-Seeger, including the case where the Hausdorff and Fourier dimension do not coincide.
Failure of weak-type endpoint restriction estimates for quadratic manifolds
本文通过构造可被少量δ-球覆盖且包含所有垂直于二次流形的平面平移的集合，证明了n维二次流形的傅里叶延拓算子在限制端点处不可能具有弱型有界性。
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.
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.
Besicovitch Compression and a Quantitative Failure of Endpoint Fourier Restriction
该研究将 Besicovitch 压缩转化为端点 Fourier 限制的定量失效，构造显式集合序列，以精确速率展示球面和抛物面受限弱型估计的退化，并指出对高维矩曲线不适用。
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.
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.
The $L^2 \to L^p$ adjoint Fourier restriction inequality on the $d$-dimensional hyperboloid $\mathbb{H}^d \subset \mathbb{R}^{d+1}$ holds provided $6 \leq p < \infty$, if $d=1$, and $2(d+2)/d \leq p\leq 2(d+1)/(d-1)$, if $d\geq2$. Quilodrán recently found the values of the optimal constants in the endpoint cases $(d,p)\in\{(2,4),(2,6),(3,4)\}$ and showed that the inequality does not have extremizers in these cases. In this paper we answer two questions posed by Quilodrán, namely: (i) we find the explicit value of the optimal constant in the endpoint case $(d,p) = (1,6)$ (the remaining endpoint…
We prove $L^p \rightarrow L^q$ Fourier restriction estimates for 3-dimensional quadratic surfaces in $\mathbb{R}^5$. Our results are sharp, up to endpoints, for a few classes of surfaces.
We prove the existence of functions that extremize the endpoint $L^2$ to $L^4$ adjoint Fourier restriction inequality on the one-sheeted hyperboloid in Euclidean space $\mathbb{R}^4$ and that, taking symmetries into consideration, any extremizing sequence has a subsequence that converges to an extremizer.
In this paper we establish the optimal multilinear restriction estimate for n-1 hypersurfaces with some curvature, where $n$ is the dimension of the underlying space. The result is sharp up to the endpoint and the role of curvature is made precise in terms of the shape operator.
Sharp L^4 restriction estimates for surfaces of codimension two in R^5
研究R^5中三维二次曲面的L^p→L^4傅里叶伴随限制估计，证明其在端点处精确，改进了前人结果。
We prove a couple of new endpoint geodesic restriction estimates for eigenfunctions. In the case of general 3-dimensional compact manifolds, after a $TT^*$ argument, simply by using the $L^2$-boundedness of the Hilbert transform on $\R$, we are able to improve the corresponding $L^2$-restriction bounds of Burq, Gérard and Tzvetkov and Hu. Also, in the case of 2-dimensional compact manifolds with nonpositive curvature, we obtain improved $L^4$-estimates for restrictions to geodesics, which, by Hölder's inequality and interpolation, implies improved $L^p$-bounds for all exponents $p\ge 2$. We do…
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 theorem for compact subsets of a type $k$ conical surface, up to an endpoint. Furthermore, the chosen weight is shown to be, in some quantitative sense, optimal. Appended is a discussion of type k conical restriction theorems which addresses some anomalies present in the existing literature.
We prove sharp endpoint results for the Fourier restriction operator associated to nondegenerate curves in $\Bbb R^d$, $d\ge 3$, and related estimates for oscillatory integral operators. Moreover, for some larger classes of curves in $\Bbb R^d$ we obtain sharp uniform $L^p\to L^q$ bounds with respect to affine arclength measure, thereby resolving a problem of Drury and Marshall.
Endpoint eigenfunction restriction estimates in codimension two
本文研究光滑闭黎曼流形上余维二子流形的拉普拉斯特征函数端点L²限制估计，证明了经典估计的little-oh改进并构造反例表明其最优性，同时否定对数移除问题。
In a recent paper [Ann. of Math. 189 (2019), 837--861], Du and Zhang proved a fractal Fourier restriction estimate and used it to establish the sharp $L^2$ estimate on the Schrödinger maximal function in $\Bbb R^n$, $n \geq 2$. In this paper, we show that the Du-Zhang estimate is the endpoint of a family of fractal restriction estimates such that each member of the family (other than the original) implies a sharp Kakeya result in $\Bbb R^n$ that is closely related to the polynomial Wolff axioms. We also prove that all the estimates of our family are true in $\Bbb R^2$.
For any $α\in(0,d)$, we construct Cantor sets in $\mathbb{R}^d$ of Hausdorff dimension $α$ such that the associated natural measure $μ$ obeys the restriction estimate $\| \widehat{f dμ} \|_{p} \leq C_p \| f \|_{L^2(μ)}$ for all $p>2d/α$. This range is optimal except for the endpoint. This extends the earlier work of Chen-Seeger and Shmerkin-Suomala, where a similar result was obtained by different methods for $α=d/k$ with $k\in\mathbb{N}$. Our proof is based on the decoupling techniques of Bourgain-Demeter and a theorem of Bourgain on the existence of $Λ(p)$ sets.
On the finite field spherical restriction conjecture in four dimensions: the sharp endpoint and applications
本文引入水平切片和平面-直线停止时间分解方法，通过将每个临界水平切片分解为富平面、富直线-贫平面和贫直线-贫平面分量，克服了球面Bochner-Riesz核中的Kloosterman障碍，证明了对于每个非零球面S_j和每个r>23/7，有R_{S_j}^*(2→r)≲1，从而在四维素数域Erdős-Falconer距离问题中首次改进了二十年前的(d+1)/2阈值。
We prove that the finite field Fourier extension operator for the paraboloid is bounded from $L^2\to L^r$ for $r\geq \frac{2d+4}{d}$ in even dimensions $d\ge 8$, which is the optimal $L^2$ estimate. For $d=6$ we obtain the optimal range $r> \frac{2d+4}{d}=8/3$, apart from the endpoint. For $d=4$ we improve the prior range of $r>16/5=3.2$ to $r\geq 28/9=3.111\ldots$, compared to the conjectured range of $r\geq3$. The key new ingredient is improved additive energy estimates for subsets of the paraboloid.
We study $L^p\to L^r$ estimates for restricted averaging operators related to algebraic varieties $V$ of $d$-dimensional vector spaces over finite fields $\mathbb F_q$ with $q$ elements. We observe properties of both the Fourier restriction operator and the averaging operator over $V\subset \mathbb F_q^d.$ As a consequence, we obtain optimal results on the restricted averaging problems for spheres and paraboloids in dimensions $d\ge2,$ and cones in odd dimensions $d\ge 3.$ In addition, when the variety $V$ is a cone lying in an even dimensional vector space over $\mathbb F_q$ and $-1$ is a squ…
Fourier Restriction Beyond Dimension Data: A Multiscale Moment Criterion
该研究针对加权 Moran 测度建立超越维数数据的傅里叶限制精确准则，揭示最优限制范围不由标准傅里叶或维数数据决定，还构造了具有相同部分属性但临界限制指数不同的测度。
It is well known that the Stein-Tomas $L^2$ Fourier restriction theorem can be used to derive sharp $L^p$ bounds for radial Fourier multipliers such as the Bochner-Riesz means. In a similar manner, $L^p \to L^2$ estimates for spectral projection operators have been utilized in order to obtain sharp $L^p$ bounds for spectral multipliers of self-adjoint elliptic pseudo-differential operators on compact manifolds. In this paper, we refine an endpoint result for spectral multipliers due to Seeger, providing endpoint bounds in terms of Besov spaces. Our proof is based on the ideas from the recent w…
Endpoint Estimates for Commutators of the Dirichlet Riesz Transform on the Half-Space
本文研究上半空间 Dirichlet Riesz 变换交换子的端点有界性，证明其从 Hardy 空间到弱 L1 有界，并给出强端点刻画：交换子有界当且仅当符号属于对数 BMO 空间，该条件反映边界距离，体现 Dirichlet 几何特性。
Fourier restriction estimates based on $L^q$-dimensions: beyond Stein--Tomas
本文提出一种新的傅里叶限制定理，用 $L^q$ 维数替代 Frostman 条件，得到连续范围的估计，在端点恢复 Stein--Tomas 结果，并部分解决 Bak 和 Seeger 的问题。
Dimension-free Fourier restriction inequalities
该研究探究高维下球面傅里叶限制不等式算子范数的渐近行为，建立径向函数的无维数端点Stein-Tomas不等式与一般函数的$O(d^{1/2})$估计，方法基于贝塞尔函数渐近估计的一致双侧改进。
Near-optimal restriction estimates for Cantor sets on the parabola
本文在抛物线上构造了维数为α的Salem康托集，证明其相关测度的Fourier变换满足L^p估计，且p>6/α的范围除端点外最优，方法结合了随机康托集限制估计与Fourier衰减技术。
A new approach to the Fourier extension problem for the paraboloid
本文提出基于二次调制波包离散化的新方法研究 Fourier 限制猜想，证明了全张量假设下直至端点的多线性延拓估计成立，并推广至弱横截性与非 L2 输入情形，验证了相关猜想特例。
Restriction of Fractional Derivatives of the Fourier Transform
本文将傅里叶变换的分数阶导数限制到非零高斯曲率曲面上，在给定曲面限制正则性条件下建立其负阶 Sobolev 空间归属，推广了 GoldStol 的定理。
Multi-parameter maximal Fourier restriction
该研究将先验傅里叶限制估计强化为多参数极大估计，推导了多参数勒贝格点性质，建立了d≥2时的d参数Menshov–Paley–Zygmund型定理，并拓展了Strichartz估计的伪微分形式。
The Stein-Tomas inequality under the effect of symmetries
该研究在O(d-k)×O(k)对称函数类上得到了单位球面的新傅里叶限制估计，证明了端点Stein-Tomas不等式在此类中极大值元的存在性，并通过构造例子验证了指数范围的最优性。
$L^{p}$-integrability of functions with Fourier supports on fractal sets on the moment curve
本研究针对矩曲线上满足特定覆盖数条件的分形紧集$E$，证明了傅里叶支撑在$E$上的$L^p$函数在特定$p$范围内恒为零，验证了$p$范围的最优性，推广了已有结果并给出相关应用。
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