Maximal operators and Fourier restriction on the moment curve

Fourier restriction to smooth enough curves

We prove Fourier restriction estimates to arbitrary compact $C^{N}$ curves for any $N > d$ in the (sharp) Drury range, using a power of the affine arclength measure as a mitigating factor. In particular, we make no nondegeneracy assumption on the curve.

Global Maximizers for the Sphere Adjoint Fourier Restriction Inequality

We show that constant functions are global maximizers for the adjoint Fourier restriction inequality for the sphere.

Large sets without Fourier restriction theorems

We construct a function that lies in $L^p(\mathbb{R}^d)$ for every $p \in (1,\infty]$ and whose Fourier transform has no Lebesgue points in a Cantor set of full Hausdorff dimension. We apply Kovač's maximal restriction principle to show that the same full-dimensional set is avoided by any Borel measure satisfying a nontrivial Fourier restriction theorem. As a consequence of a near-optimal fractal restriction theorem of Łaba and Wang, we hence prove a lack of valid relations between the Hausdorff dimension of a set and the range of possible Fourier restriction exponents for measures supported i…

A sharp Lp Lq Fourier restriction theorem for a conical surface of finite type

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.

Application of a Fourier restriction theorem to certain families of projections in R^3

We use a restriction theorem for Fourier transforms of fractal measures to study projections onto families of planes in R^3 whose normal directions form nondegenerate curves.

Endpoint Fourier restriction and unrectifiability

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.

Fourier restriction estimates for surfaces of co-dimension two in $\mathbb{R}^5$

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.

Bilinear Fourier restriction theorems

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 approaches, our argument avoids any use of the Rubio de Francia Littlewood--Paley inequality.

Low-dimensional maximal restriction principles for the Fourier transform

Following the ideas from a paper by the same author, we prove abstract maximal restriction results for the Fourier transform. Our results deal mainly with maximal operators of convolution-type and $r-$average maximal functions. As a by-product of our techniques we obtain spherical maximal restriction estimates, as well as restriction estimates for $2-$average maximal functions, answering thus points left open by V. Kovač and Müller, Ricci and Wright.

$L^p-L^2$ Fourier restriction for hypersurfaces in $\Bbb R^3$: Part II

This is the second of two articles in which we prove a sharp $L^p-L^2$ Fourier restriction theorem for a large class of smooth, finite type hypersurfaces in R^3, which includes in particular all real-analytic hypersurfaces.

Existence of extremals for a Fourier restriction inequality on the one-sheeted hyperboloid

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.

Extremizability of Fourier restriction to the paraboloid

In this article, we prove that all global, nonendpoint Fourier restriction inequalities for the paraboloid in $\mathbb R^{1+d}$ have extremizers and that $L^p$-normalized extremizing sequences are precompact modulo symmetries. This result had previously been established for the case $q=2$. In the range where the boundedness of the restriction operator is still an open question, our result is conditional on improvements toward the restriction conjecture.

Fourier transform, $L^2$ restriction theorem, and scaling

We show, using a Knapp-type homogeneity argument, that the $(L^p, L^2)$ restriction theorem implies a growth condition on the hypersurface in question. We further use this result to show that the optimal $(L^p, L^2)$ restriction theorem implies the sharp isotropic decay rate for the Fourier transform of the Lebesgue measure carried by compact convex finite hypersurfaces.

The Fourier restriction and Kakeya problems over rings of integers modulo $N$

The Fourier restriction phenomenon and the size of Kakeya sets are explored in the setting of the ring of integers modulo $N$ for general $N$ and a striking similarity with the corresponding euclidean problems is observed. One should contrast this with known results in the finite field setting.

Improved estimates for the discrete Fourier restriction to the higher dimensional sphere

We improve the exponent for the discrete Fourier restriction to the $n$ dimensional sphere, from $p=\frac{2(n+1)}{n-3}$ to $p=\frac{2n}{n-3}$, when $n\ge 4$.

Sets of Salem type and sharpness of the $L^2$-Fourier restriction theorem

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.

Gaussians Rarely Extremize Adjoint Fourier Restriction Inequalities For Paraboloids

It was proved independently by Foschi and Hundertmark, Zharnitsky that Gaussians extremize the adjoint Fourier restriction inequality for L^2 functions on the paraboloid in the two lowest-dimesional cases. Here we prove that Gaussians are critical points for the L^p to L^q adjoint Fourier restriction inequalities if and only if p=2. Also, Gaussians are critial points for the L^2 to L^r_t L^q_x Strichartz inequalities for all admissible pairs (r,q) in (1,infinity)^2.

Uniform estimates for Fourier restriction to polynomial curves in $\mathbb R^d$

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.

An affine Fourier restriction theorem for conical surfaces

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.

缓慢衰减的拉贾克曼测度与傅里叶变换在零傅里叶维度极限情况下的限制定理

Slowly decaying Rajchman measures and a restriction theorem for the Fourier transform at the limit case of zero Fourier dimension

本文证明了存在零傅里叶维度的集合，使得傅里叶变换可以在特定非平凡区间[1,~p)上限制，1<~p<2。该研究基于Mockenhaupt的限制定理，结合多项对数傅里叶衰减的测度和全Hausdorff维度1的集合，扩展了限制定理的应用范围。

Radial multipliers and restriction to surfaces of the Fourier transform in mixed-norm spaces

In this article we revisit some classical conjectures in harmonic analysis in the setting of mixed norm spaces $L^p_{rad} L^2_{ang} (\mathbb{R}^n)$. We produce sharp bounds for the restriction of the Fourier transform to compact hypersurfaces of revolution in the mixed norm setting and study an extension of the disc multiplier. We also present some results for the discrete restriction conjecture and state an intriguing open problem.

A Fourier restriction estimate for surfaces of positive curvature in R^6

We improve the best known exponent for the restriction conjecture in R^6. Our idea is applicable to any dimension n satisfying n = 0 mod 3, though we do not explicitly calculate the improvement for n > 6. This improves the recent results of Bourgain and Guth.

Existence of extremizers for Fourier restriction to the moment curve

We show that the restriction and extension operators associated to the moment curve possess extremizers and that $L^p$-normalized extremizing sequences of these operators are precompact modulo symmetries.

Scale invariant Fourier restriction to a hyperbolic surface

This result sharpens the bilinear to linear deduction of Lee and Vargas for extension estimates on the hyperbolic paraboloid in $\mathbb R^3$ to the sharp line, leading to the first scale-invariant restriction estimates, beyond the Stein--Tomas range, for a hypersurface on which the principal curvatures have different signs.

Improved Fourier restriction estimates in higher dimensions

We consider Guth's approach to the Fourier restriction problem via polynomial partitioning. By writing out his induction argument as a recursive algorithm and introducing new geometric information, known as the polynomial Wolff axioms, we obtain improved bounds for the restriction conjecture, particularly in high dimensions. Consequences for the Kakeya conjecture are also considered.

Maximal restriction estimates and the maximal function of the Fourier transform

We prove a maximal restriction inequality for the Fourier transform, providing an answer to a question left open by Müller, Ricci and Wright. Our methods are similar to the ones in their article, with the addition of a suitable trick to help us linearise our maximal function. In the end, we comment on how to use the same trick in combination with Vitturi's approach to obtain a stronger high-dimensional result.

Extremizers for adjoint Fourier restriction on hyperboloids: the higher dimensional case

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.

Fourier restriction implies maximal and variational Fourier restriction

We give an abstract argument that an a priori Fourier restriction estimate for a certain choice of exponents automatically implies maximal and variational Fourier restriction estimates. These, in turn, provide pointwise and quantitative interpretations of restriction of the Fourier transform to the hypersurface in question.

关于傅里叶限制与嵌套多项式沃尔夫公理的注记

A note on Fourier restriction and nested Polynomial Wolff axioms

本文通过结合多项式划分方法与几何结果，改进了高维空间中傅里叶限制猜想的L^p范围。

A Fourier Restriction Theorem For A Twodimensional Surface Of Finite Type

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 the study of more general $L^p(R^3)\to L^q(S)$ Fourier restriction estimates, by studying a prototypical class of two-dimensional surfaces with strongly varying curvature conditions. Our approach is based on an adaptation of the so-called bilinear method. We discuss several new features arising in the study of this problem.

Fourier restriction for smooth hyperbolic 2-surfaces

We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in threedimensional euclidean space. Our approach exploits in a crucial way the underlying hyperbolic geometry, which leads to a novel notion of strong transversality and corresponding "exceptional" sets. For the division of these exceptional sets we make crucial and perhaps surprising use of a lemma on level sets for sufficiently smooth one-variate functions from a previous article of ours.

Some recent progress on sharp Fourier restriction theory

The purpose of this note is to discuss several results that have been obtained in the last decade in the context of sharp adjoint Fourier restriction/Strichartz inequalities. Rather than aiming at full generality, we focus on several concrete examples of underlying manifolds with large groups of symmetries, which sometimes allows for simple geometric proofs. We mention several open problems along the way, and include an appendix on integration on manifolds using delta calculus.

Discrete Fourier restriction theorems in two dimensions

Consider the group ${\mathbb{R}}^2$ with the discrete topology, and denote its Fourier algebra by $A({\mathbb{R}_{\rm d}^2})$. We reformulate a theorem of V.A. Yudin as a statement about restrictions of functions in $A({\mathbb{R}_{\rm d}^2})$ to the boundary of a strictly convex domain when those functions vanish outside that boundary. We give visual proofs of that statement and a complementary one.

Explicit Salem sets, Fourier restriction, and metric Diophantine approximation in the $p$-adic numbers

We exhibit the first explicit examples of Salem sets in $\mathbb{Q}_p$ of every dimension $0 < α< 1$ by showing that certain sets of well-approximable $p$-adic numbers are Salem sets. We construct measures supported on these sets that satisfy essentially optimal Fourier decay and upper regularity conditions, and we observe that these conditions imply that the measures satisfy strong Fourier restriction inequalities. We also partially generalize our results to higher dimensions. Our results extend theorems of Kaufman, Papadimitropoulos, and Hambrook from the real to the $p$-adic setting.

Uniform estimates for the Fourier transform of surface carried measures in $\Bbb R^3$ and an application to Fourier restriction

Let $S$ be a hypersurface in $\Bbb R^3$ which is the graph of a smooth, finite type function $ϕ,$ and let $μ=ρ\, d\si$ be a surface carried measure on $S,$ where $d\si$ denotes the surface element on $S$ and $ρ$ a smooth density with suffiently small support. We derive uniform estimates for the Fourier transform $\hat μ$ of $μ,$ which are sharp except for the case where the principal face of the Newton polyhedron of $ϕ,$ when expressed in adapted coordinates, is unbounded. As an application, we prove a sharp $L^p$-$L^2$ Fourier restriction theorem for $S$ in the case where the original coordin…

Restriction of Fourier transforms to curves, II: Some classes with vanishing torsion

We consider the Fourier restriction operators associated to certain degenerate curves in R^d for which the highest torsion vanishes. We prove estimates with respect to affine arclength and with respect to the Euclidean arclength measure on the curve. The estimates have certain uniform features, and the affine arclength results cover families of flat curves.

A Fourier restriction theorem for a perturbed hyperbolic paraboloid: polynomial partitioning

We consider a surface with negative curvature in $\Bbb R^3$ which is a cubic perturbation of the saddle. For this surface, we prove a new restriction theorem, analogous to the theorem for paraboloids proved by L. Guth in 2016. This specific perturbation has turned out to be of fundamental importance also to the understanding of more general classes of perturbations.

空间曲线傅里叶限制在正交函数系上的研究

Space curve Fourier restriction for orthonormal functions

本文通过希尔伯特空间方法和高低方法变体，证明了正交函数系下非退化曲线的$L^2$傅里叶限制不等式，并建立了相关卷积算子的$L^p$改进界。

Restriction of Fourier transforms to curves and related oscillatory integrals

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.

A note on maximal Fourier Restriction for spheres in all dimensions

We prove a maximal Fourier restriction theorem for the sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^{d}$ for any dimension $d\geq 3$ in a restricted range of exponents given by the Stein-Tomas theorem. The proof consists of a simple observation. When $d=3$ the range corresponds exactly to the full Stein-Tomas one, but is otherwise a proper subset when $d>3$. We also present an application regarding the Lebesgue points of functions in $\mathcal{F}(L^p)$ when $p$ is sufficiently close to 1.

$L^p$加权傅里叶限制估计

$L^p$ weighted Fourier restriction estimates

该研究在数学调和分析领域，针对截顶抛物面上的傅里叶延拓算子，得到了带特定$n$维权函数的精确$L^p$加权傅里叶限制估计式。

A family of fractal Fourier restriction estimates with implications on the Kakeya problem

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 $\Bbb R^2$.

单项曲线的尖锐傅里叶限制

Sharp Fourier restriction to monomial curves

本文建立了带有仿射弧长测度的单项曲线的傅里叶限制/延拓算子范数的下界，并证明了极值化序列预紧性与范数严格大于阈值的等价关系，同时引入了适用于更一般流形的新方法。

A sharp trilinear inequality related to Fourier restriction on the circle

In this paper we prove a sharp trilinear inequality which is motivated by a program to obtain the sharp form of the $L^2 - L^6$ Tomas-Stein adjoint restriction inequality on the circle. Our method uses intricate estimates for integrals of sixfold products of Bessel functions developed in a companion paper. We also establish that constants are local extremizers of the Tomas-Stein adjoint restriction inequality as well as of another inequality appearing in the program.

On Fourier restriction for finite-type perturbations of the hyperboloid

In this note, we continue our research on Fourier restriction for hyperbolic surfaces, by studying local perturbations of the hyperbolic paraboloid $z=xy,$ which are of the form $z=xy+h(y),$ where $h(y)$ is a smooth function of finite type. Our results build on previous joint work in which we have studied the case $h(y)=y^3/3$ by means of the bilinear method. As it turns out, the understanding of that special case becomes also crucial for the treatment of arbitrary finite type perturbation terms $h(y).$

A geometric lemma for complex polynomial curves with applications in Fourier restriction theory

The aim of this paper is to prove a uniform Fourier restriction estimate for certain $2-$dimensional surfaces in $\mathbb R^{2n}$. These surfaces are the image of complex polynomial curves $γ(z) = (p_1(z), \dots, p_n(z))$, equipped with the complex equivalent to the affine arclength measure. This result is a complex-polynomial counterpart to a previous result by Stovall [Sto16] in the real setting. As a means to prove this theorem we provide an alternative proof of a geometric inequality by Dendrinos and Wright [DW10] that extends the result to complex polynomials.

伪微分算子的界与 Schatten 类的 Fourier 限制

Bounds on pseudodifferential operators and Fourier restriction for Schatten classes

本文证明了具有紧 Fourier 支撑的 Weyl 伪微分算子属于 Schatten 类当且仅当其符号属于相空间的 $L^p$ 空间，并以此提供了 Luef 和 Samuelsen 关于经典与量子限制估计等价性的清晰证明。

A maximal restriction theorem and Lebesgue points of functions in F(L^p)

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 given subvariety S, endowed with a suitabel measure. Such estimates allow to define the restriction Rf of the Fourier transform of an L^p-function to S in an operator theoretic sense. In this article, we begin to investigate the question what is the "intrinsic" pointwise relation between Rf and the Fourier transform of f, by looking at curves in the plane, for…

On extremizing sequences for adjoint Fourier restriction to the sphere

In this article, we develop a linear profile decomposition for the $L^p \to L^q$ adjoint Fourier restriction operator associated to the sphere, valid for exponent pairs $p<q$ for which this operator is bounded. Such theorems are new when $p \neq 2$. We apply these methods to prove new results regarding the existence of extremizers and the behavior of extremizing sequences for the spherical extension operator. Namely, assuming boundedness, extremizers exist if $q>\max\{p,\tfrac{d+2}d p'\}$, or if $q=\tfrac{d+2}d p'$ and the operator norm exceeds a certain constant times the operator norm of the…

A Cotlar Type Maximal Function Associated With Fourier Multipliers

We prove the $L^p$ boundedness of a maximal operator associated with a dyadic frequency decomposition of a Fourier multiplier, under a weak regularity assumption.

$\mathbb{R}^3$ 中光滑曲面的 $L^2$ 仿射 Fourier 限制定理

$L^2$ affine Fourier restriction theorems for smooth surfaces in $\mathbb{R}^3$

本文证明了 $\mathbb{R}^3$ 中光滑曲面上尖锐的 $L^2$ 仿射 Fourier 限制不等式，适用于所有光滑曲面，并利用解耦定理获得对多项式图像曲面的统一界。

凸曲线的一致傅里叶限制

Uniform Fourier restriction for convex curves

本研究推广了极大傅里叶限制算子估计至平面任意凸曲线，去掉了原有结果的$\u2702$正则性条件，借助仿射不变构造选取适配测度，得到一致傅里叶限制定理及曲线上傅里叶变换勒贝格点相关结果。

Schatten类算子与相空间上函数的傅里叶限制

Fourier restriction for Schatten class operators and functions on phase space

该研究提出Schatten类算子的Fourier-Wigner傅里叶限制变体，证明其等价于相空间函数的辛傅里叶变换限制，推导了量子傅里叶延拓算子的Schatten类结果，肯定证明了Mishra与Vemuri关于测度Weyl变换的猜想。

New bounds for the discrete Fourier restriction to the sphere in four and five dimensions

We improve the range for the discrete Fourier restriction to the four and five dimensional spheres. We rely on two new ingredients, incidence theory and Siegel's mass formula.