Richard Oberlin
A (d,k) set is a subset of R^d containing a translate of every k-dimensional plane. Bourgain showed that for 2^{k-1}+k \geq d, every (d,k) set has positive Lebesgue measure. We give an L^p bound for the corresponding maximal operator.
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We shall verify the Kakeya (Nikodym) maximal operator $K_{N}$, $N\gg 1$, is bounded on the variable Lebesgue space $L^{p(\cdot)}(\mathbb{R}^2)$ when the exponent function $p(\cdot)$ is $N$-modified locally log-Hölder continuous and log-Hölder continuous at infinity.
A Kakeya maximal estimate for regulus strips
研究扭歪直线带的卡克亚型估计，通过证明其不相交对卡克亚猜想有改进，还建立了第一海森堡群中尼科迪姆型极大函数的\(L^p\)不等式。
In this paper, we will introduce and study several types of Kakeya inequalities by the maximal functions in Hardy spaces in $\RR^n$,\,$(n\geq2)$, and we could obtain several inequalities associated with the Kakeya inequalities. We will show that $\big\|M^t_{δS_{α,β}} f \big\|_p\lesssim_{p,n,φ,\varepsilon}\left(\frac{1}δ\right)^{5\left(\frac{n}{r}+2\right)\varepsilon}\big\|f\big\|_p$, when $f(x)\in L^p(\RR^n)$ and $supp\,\hat{f}(ξ)\subseteq B(0,1)$.
Equivalence of linear and trilinear Kakeya conjectures in three dimensions
本文证明三维中 Kakeya 极大算子猜想与其不相交三线性对偶形式彼此等价。
In a prior work [Hilbert transform along smooth families of lines math.CA/0310345] the authors introduced a variant of the Kakeya maximal function associated with Lipschitz maps from the plane into the unit circle. In this paper, we improve the known estimates for this maximal operator--and raise the conjecture that the bounds established are optimal.
Bilinear Bochner-Riesz means for convex domains and Kakeya Maximal function
该研究引入平面凸域相关的双线性Bochner-Riesz平均，借助双线性Kakeya极大函数建立其$L^p$估计以证明有界性，并揭示该极大函数与Lacey双线性极大函数的联系。
We develop a notion of finite order lacunarity for direction sets in $\mathbb R^{d+1}$. Given a direction set $Ω$ that is sublacunary according to this definition, we construct random examples of Euclidean sets that contain unit line segments with directions from $Ω$ and enjoy analytical features similar to those of traditional Kakeya sets of infinitesimal Lebesgue measure. This generalizes to higher dimensions a planar result due to Bateman. Combined with earlier work of Alfonseca, Bateman, Parcet and Rogers, this notion of lacunarity and Kakeya-type sets also yields a characterization in all…
We prove that the maximal operator obtained by taking averages at scale 1 along $N$ arbitrary directions on the sphere, is bounded in $L^2(\R^3)$ by $N^{1/4}{\log N}$. When the directions are $N^{-1/2}$ separated, we improve the bound to $N^{1/4}\sqrt{\log N}$. Apart from the logarithmic terms these bounds are optimal.
Restriction and Kakeya maximal estimates in $\mathbb{R}^4$
该研究结合平面刷论证与解耦- incidence方法，推导出R^4中Fourier限制和Bochner-Riesz问题的新界，并获得维度3.054的凯凯ya极大估计。
We show that for any dimension $d\ge3$, one can obtain Wolff's $L^{(d+2)/2}$ bound on Kakeya-Nikodym maximal function in $\mathbb R^d$ for $d\ge3$ without the induction on scales argument. The key ingredient is to reduce to a 2-dimensional $L^2$ estimate with an auxiliary maximal function. We also prove that the same $L^{(d+2)/2}$ bound holds for Nikodym maximal function for any manifold $(M^d,g)$ with constant curvature, which generalizes Sogge's results for $d=3$ to any $d\ge3$. As in the 3-dimensional case, we can handle manifolds of constant curvature due to the fact that, in this case, tw…
A (d,k) set is a subset of R^d containing a translate of every k-dimensional plane. Bourgain showed that for k \geq k_{cr}(d), where k_{cr}(d) solves 2^{k_{cr}-1}+k_{cr} = d, every (d,k) set has positive Lebesgue measure. We give a short proof of this result which allows for an improved L^p estimate of the corresponding maximal operator, and which demonstrates that a lower value of k_{cr} could be obtained if improved mixed-norm estimates for the x-ray transform were known.
Given a family G of rectangles, to which one associates a tree [G], one defines a natural number $λ$ [G] called its analytic split and satisfying, for all 1 < p < $\infty$ log($λ$ [G]) p MG p p where MG is the Hardy-Littlewood type maximal operator associated to the family G. As an application, we completely characterize the boundeness of planar rarefied directional maximal operators on L p for 1 < p < $\infty$. Precisely, if $Ω$ is an arbitrary set of angles in [0, $π$ 4), we prove that any rarefied basis B of the directional basis R $Ω$ yields an operator MB that has the same L p-behavior th…
A Kakeya maximal function estimate in four dimensions using planebrushes
本文提出平面刷这一新几何工具，在四维空间中改进了Kakeya极大函数估计，得到维度3.049的极大函数估计，并证明Besicovitch集合的Hausdorff维数至少为3.059。
We derive Maximal Kakeya estimates for functions over $\mathbb{Z}/N\mathbb{Z}$ proving the Maximal Kakeya conjecture for $\mathbb{Z}/N\mathbb{Z}$ for general $N$ as stated by Hickman and Wright [HW18]. The proof involves using polynomial method and linear algebra techniques from [Dha21, Ars21a, DD21] and generalizing a probabilistic method argument from [DD22]. As another application we give lower bounds for the size of $(m,ε)$-Kakeya sets over $\mathbb{Z}/N\mathbb{Z}$. Using these ideas we also give a new, simpler, and direct proof for Maximal Kakeya bounds over finite fields (which were firs…
We adapt Guth's polynomial partitioning argument for the Fourier restriction problem to the context of the Kakeya problem. By writing out the induction argument as a recursive algorithm, additional multiscale geometric information is made available. To take advantage of this, we prove that direction-separated tubes satisfy a multiscale version of the polynomial Wolff axioms. Altogether, this yields improved bounds for the Kakeya maximal conjecture in $\mathbb{R}^n$ with $n=5$ or $n\ge 7$ and improved bounds for the Kakeya set conjecture for an infinite sequence of dimensions.
We discuss a planar variant of the Kakeya maximal function in the setting of a vector space over a finite field. Using methods from incidence combinatorics, we demonstrate that the operator is bounded from $L^p$ to $L^q$ when $1 \leq p \leq \frac{kn+k+1}{k(k+1)}$ and $1 \leq q \leq (n-k)p'$.
We completely characterize the boundedness of planar directional maximal operators on L^p. More precisely, if Omega is a set of directions, we show that M_Omega, the maximal operator associated to line segments in the directions Omega, is unbounded on L^p, for all p < infinity, precisely when Omega admits Kakeya-type sets. In fact, we show that if Omega does not admit Kakeya sets, then Omega is a generalized lacunary set, and hence M_Omega is bounded on L^p, for p>1.
We reprove Wolff's $L^{\frac{5}2}-$ bound for the $\R^3-$Kakeya maximal function without appealing to the argument of induction on scales. The main ingredient in our proof is an adaptation of Sogge's strategy used in the work on Nikodym-type sets in curved spaces. Although the equivalence between these two type maximal functions is well known, our proof may shed light on some new geometric observations which is interesting in its own right.
In the finite field setting, we show that the restriction conjecture associated to any one of a large family of $d=2n+1$ dimensional quadratic surfaces implies the $n+1$ dimensional Kakeya conjecture (Dvir's theorem). This includes the case of the paraboloid over finite fields in which $-1$ \emph{is} a square. We are able to partially reverse this implication using the sharp Kakeya maximal operator estimates of Ellenberg, Oberlin and Tao to establish the first finite field restriction estimates beyond the Stein-Tomas exponent in this setting.
We prove that the Kakeya maximal conjecture is equivalent to the $Ω$-Kakeya maximal conjecture. This completes a recent result in [2] where Keleti and Math{é} proved that the Kakeya conjecture is equivalent to the $Ω$-Kakeya conjecture. Moreover, we improve concrete bound on the Hausdorff dimension of a $Ω$-Kakeya set : for any Bore set $Ω$ in S n--1 , we prove that if X $\subset$ R n contains for any e $\in$ $Ω$ a unit segment oriented along e then we have dX $\ge$ 6 11 d$Ω$ + 1 where dE denotes the Hausdorff dimension of a set E.
Lifting curved Kakeya sets to linear Kakeya sets
本文证明弯曲Kakeya问题可提升为高维线性Kakeya问题，若经典猜想成立，则全维曲线族在Hörmander型族中稠密，并深化了对相关问题的理解。
An $L^{4/3}$ $SL_2$ Kakeya maximal inequality
本文构造性地证明了R³中存在零测度SL₂ Besicovitch集，由此说明SL₂ Kakeya极大不等式不能以一致常数成立；进一步给出p>3/2时的反例，并在二维球条件下证明p≤4/3时该不等式在C_εδ^{-ε}损失下成立。
We obtain new bounds for the Kakeya maximal conjecture in most dimensions $n<100$, as well as improved bounds for the Kakeya set conjecture when $n=7$ or $9$. For this we consider Guth and Zahl's strengthened formulation of the maximal conjecture, concerning families of tubes that satisfy the polynomial Wolff axioms. Our results give improved estimates for this strengthened formulation when $n =5$ or $n \geq 7$.
We establish new estimates on the Minkowski and Hausdorff dimensions of Besicovitch sets and obtain new bounds on the Kakeya maximal operator.
Kakeya maximal inequality in the Heisenberg group
本文在配备 Korányi 距离的海森堡群中定义了 Kakeya 极大函数，利用 Wolff 圆形极大函数定理的变体证明了其 $L^3$ 范数不等式，并由此恢复了海森堡 Kakeya 集 Hausdorff 维度的精确下界。
I show that $L^{p}-L^{q}$ estimates for the Kakeya maximal function yield lower bounds for the conformal dimension of Kakeya sets, and upper bounds for how much quasisymmetries can increase the Hausdorff dimension of line segments inside Kakeya sets. Combining the known $L^{p}-L^{q}$ estimates of Wolff and Katz-Tao with the main result of the paper, the conformal dimension of Kakeya sets in $\mathbb{R}^{n}$ is at least $\max\{(n + 2)/2,(4n + 3)/7\}$. Moreover, if $f$ is a quasisymmetry from a Kakeya set $K \subset \mathbb{R}^{n}$ onto any at most $n$-dimensional metric space, the $f$-image of …
Given a Cantor-type subset $Ω$ of a smooth curve in $\mathbb R^{d+1}$, we construct examples of sets that contain unit line segments with directions from $Ω$ and exhibit analytical features similar to those of classical Kakeya sets of arbitrarily small $(d+1)$-dimensional Lebesgue measure. The construction is based on probabilistic methods relying on the tree structure of $Ω$, and extends to higher dimensions an analogous planar result of Bateman and Katz. In particular, the existence of such sets implies that the directional maximal operator associated with the direction set $Ω$ is unbounded …
We establish new linear and trilinear bounds for collections of tubes in $\mathbb{R}^4$ that satisfy the polynomial Wolff axioms. In brief, a collection of $δ$-tubes satisfies the Wolff axioms if not too many tubes can be contained in the $δ$-neighborhood of a plane. A collection of tubes satisfies the polynomial Wolff axioms if not too many tubes can be contained in the $δ$-neighborhood of a low degree algebraic variety. First, we prove that if a set of $δ^{-3}$ tubes in $\mathbb{R}^4$ satisfies the polynomial Wolff axioms, then the union of the tubes must have volume at least $δ^{1-1/40}$. W…
Kakeya sets from lines in $SL_2$
本文证明由满足 $ad-bc=1$ 的直线构成的 $SL_2$ Kakeya 集在 $\n\b\bR^3$ 中 Hausdorff 维数必为 3，并通过尺度归纳、扭转投影结构及 Wolff 圆极大函数变体给出新证明。
Hausdorff dimension of restricted Kakeya sets
本文研究了限制Kakeya集的Hausdorff维数，在规定线段中点属于固定集合的条件下，利用bush argument改进了维数下界，并建立了Kakeya极大函数的类似结果。
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。
Construction of a curved Kakeya set
本文构造了 $\mathbb R^2$ 中测度为零且包含所有孔径在 1 到 2 之间抛物线段的紧集，并据此改进相应极大算子的 $L^p$-$L^q$ 范数下界，同时将构造推广到满足适当曲率条件的 $C^2$ 曲线族。
Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions
本文研究三维空间中具有限制性管集合的并集体积，证明其体积接近最大，并由此推出所有Kakeya集的Minkowski和Hausdorff维数均为3。
On the Peres--Schlag orthogonal projection problem and Kakeya-type sets
研究有限域和欧几里得空间中正交投影的 Peres--Schlag 非空内部问题，通过多项式方法建立有限域上的尖锐投影结果，并在欧几里得空间中结合几何测度论与调和分析技术改进参数范围。
The Kakeya problem for codimension-two planes in $\mathbb{R}^n$
研究n≥4时ℝⁿ中满足凸Wolff公理的余维二平面Kakeya问题，证明此类δ片的并集具有几乎最大体积，是Wang和Zahl工作的高维推广，还解决了ℝ⁴中满足曲率条件的Rogers 2平面Kakeya问题。
Cellular Maximal-Density Factoring: Joint Shadings, Stable Refinements, and a Conditional Kakeya Application
针对Guth–Wang–Zahl三维Kakeya简化证明的两处缺陷，提出细胞联合极大密度分解方法修正，完成关键估计与链推导，证明对应幂为尖锐值。
We prove optimal bounds in L^2(R^2) for the maximal oper- ator obtained by taking a singular integral along N arbitrary directions in the plane. We also give a new proof for the optimal L^2 bound for the single scale Kakeya maximal function.
Probabilistic Construction of Kakeya-Type Sets in $\mathbb{R}^2$ associated to separated sets of directions
本文给出方向集Ω的一个条件，保证其方向极大算子M_Ω在所有1≤p<∞的L^p(R^2)上无界；证明扩展Bateman-Katz方法，利用粘性映射与Bernoulli渗流概率构造Kakeya型集合。
Directional maximal operators in the plane
本研究针对平面方向极大算子$D_{\Omega}$的Lebesgue有界性问题，修正了Bateman 2009年相关证明的漏洞，引入可容许有限阶缺隙性概念，建立了有限阶缺隙性、无Kakeya型集合与算子有界性的等价刻画。
this paper has been withdrawn. A crucial estimate on a Lipschitz variant of the Kakeya maximal function has an incomplete proof.
P-adic Curved Kakeya Sets, Projection Theorem and Covering Numbers
本文证明了 p-adic 域上弯曲 Kakeya 集的 Hausdorff 维数下界，并建立了投影定理与覆盖数比较，核心方法为多项式到线提升映射与素数幂 Kakeya 估计。
On Maximal Functions Associated to Families of Curves in the Plane
本文提出统一框架研究平面曲线族极大函数，证明尖锐L^p算子界，并应用于受限投影和Furstenberg型集合维数问题。
In 1901, Severi proved that if $Z$ is an irreducible hypersurface in $\mathbb{P}^4(\mathbb{C})$ that contains a three dimensional set of lines, then $Z$ is either a quadratic hypersurface or a scroll of planes. We prove a discretized version of this result for hypersurfaces in $\mathbb{R}^4$. As an application, we prove that at most $δ^{-2-\varepsilon}$ direction-separated $δ$-tubes can be contained in the $δ$-neighborhood of a low-degree hypersurface in $\mathbb{R}^4$. This result leads to improved bounds on the restriction and Kakeya problems in $\mathbb{R}^4$. Combined with previous work of…
Horizontal Kakeya maximal operators in finite Heisenberg groups: Exact exponents and applications
研究有限Heisenberg群中与水平线相关的Kakeya极大算子，通过傅里叶分析得到秩一时精确的混合范数增长指数，并应用于组合几何中的点线配置问题。
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。
We prove L2 x L2 to weak L1 estimates for some novel bilinear maximal operators of Kakeya and lacunary type thus extending to this setting, the works of Cordoba and of Nagel, Stein and Wainger.
Oscillatory integral operators on manifolds and related Kakeya and Nikodym problems
本文研究了黎曼流形上的Carleson-Sjölin算子，在常截面曲率和Sogge混沌曲率条件下通过不同方法获得了改进的$L^p$界，并作为副产品得到了Nikodym极大函数的新界。
We show that, for $n\geq 3$, $\lim_{t \to 0} e^{itΔ}f(x) = f(x)$ holds almost everywhere for all $f \in H^s (\mathbb{R}^n)$ provided that $s>\frac{n}{2(n+1)}$. Due to a counterexample by Bourgain, up to the endpoint, this result is sharp and fully resolves a problem raised by Carleson. Our main theorem is a fractal $L^2$ restriction estimate, which also gives improved results on the size of divergence set of Schrödinger solutions, the Falconer distance set problem and the spherical average Fourier decay rates of fractal measures. The key ingredients of the proof include multilinear Kakeya esti…
Maximal directional operators along algebraic varieties
本文建立了沿任意维数多项式簇上有限方向集的极大方向平均算子的 $L^2$ 范数的尖锐增长阶，推广了 Córdoba、Katz 和 Demeter 的工作，并改进了球面情形的已知界，同时给出 Kakeya 型极大函数的新估计。
Maximal subspace averages
本文系统研究格拉斯曼流形上最大子空间平均算子，证明其L²界本质尖锐，并推广Nikodym猜想，指出d>1时Kakeya与Nikodym问题不等价。
Oscillatory integral operators and variable Schrödinger propagators: beyond the universal estimates
该研究针对实解析相位的Hörmander型振荡积分算子，推导出超越普适范围的Lp界条件，并揭示变系数薛定谔传播子的差异，核心创新在于结合多项式方法与子水平集估计证明弯曲Kakeya/Nikodym极大函数估计。
$(n,k)$-Besicovitch sets do not exist in $\mathbb{Z}_p^n$ and $\hat{\mathbb{Z}}^n$ for $k\ge 2$
本文证明了在 $p$-进数和射有限整数空间中，任何包含所有方向 $2$-平坦的集合必具有正测度，从而否定了相应 Besicovitch 型零测度构造的存在性。
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