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 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 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 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 $1 \leq q \leq (n-k)p'$.
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 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 (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.
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 $supp\,\hat{f}(ξ)\subseteq B(0,1)$.
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.
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.
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…
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.
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.
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…
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.
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.
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 $n \geq 7$.
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 construct a union of N parallelograms of dimensions approximately 1/N x 1 in the plane, with the slope of their long sides in the standard Cantor set. The union has area 1/log N but the union of the doubles has area log log N/ log N. In particular, this implies unbounded of the associated maximal operator in L^p for any p different from infinity. The construction is by randomizing an earlier construction of the second author for the L^2 case. The proof that the construction satisfies the desired conditions is by elementary estimates in the theory of percolation on trees as developed by R. L…
Given a discrete group $\G$ and an orthogonal action $γ: \G \to O(n)$ we study $L_p$ convergence of Fourier integrals which are frequency supported on the semidirect product $\R^n \rtimes_γ\G$. Given a unit $u \in \R^n$ and $1 < p \neq 2 < \infty$, our main result shows that the twisted (directional) Hilbert transform $H_u \rtimes_γid_\G$ is $L_p$-bounded iff the orbit $\mathcal{O}_γ(u)$ is finite. This is in sharp contrast with twisted Riesz transforms $R_u \rtimes_γid_\G$, which are always bounded. Our result characterizes Fourier summability in $L_p$ for this class of groups. We also extend…
Horizontal Kakeya maximal operators in finite Heisenberg groups: Exact exponents and applications
Institute of Mathematics and Interdisciplinary Sciences, Xidian University（西安电子科技大学数学与交叉科学研究院）；Department of Mathematics, University of Trento（特伦托大学数学系）；VNU University of Science, Hanoi, Vietnam（越南国立大学河内大学）；Institute of Mathematical Sciences, ShanghaiTech University（上海科技大学数理学院）
The Kakeya problem for codimension-two planes in $\mathbb{R}^n$
San Francisco State University（旧金山州立大学）；University of Pennsylvania（宾夕法尼亚大学）；Georgia Institute of Technology（佐治亚理工学院）；University of British Columbia（不列颠哥伦比亚大学）
Cellular Maximal-Density Factoring: Joint Shadings, Stable Refinements, and a Conditional Kakeya Application
Changkong College, Nanjing University of Aeronautics and Astronautics（南京航空航天大学长空学院）；School of Science, Nanjing University of Posts and Telecommunications（南京邮电大学理学院）
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.
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…
The Fefferman-Stein type inequalities for strong maximal operator and directional maximal operator are verified with composition of the Hardy-Littlewood maximal operator in the plane.
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.
Let $V = \{ v_1,\dots,v_N\}$ be a collection of $N$ vectors that live near a discrete sphere. We consider discrete directional maximal functions on $\mathbb{Z}^2$ where the set of directions lies in $V$, given by \[ \sup_{v \in V, k \geq C \log N} \left| \sum_{n \in \mathbb{Z}} f(x-v\cdot n ) \cdot ϕ_k(n) \right|, \ f:\mathbb{Z}^2 \to \mathbb{C}, \] where and $ϕ_k(t) := 2^{-k} ϕ(2^{-k} t)$ for some bump function $ϕ$. Interestingly, the study of these operators leads one to consider an "arithmetic version" of a Kakeya-type problem in the plane, which we approach using a combination of geometric…
We study a two-dimensional discrete directional maximal operator along the set of the prime numbers. We show existence of a set of vectors, which are lattice points in a sufficiently large annulus, for which the $\ell^2$ norm of the associated maximal operator with supremum taken over all large scales grows with an epsilon power in the number of vectors. This paper is a follow-up to a prior work on the discrete directional maximal operator along the integers by the first and third author.
We show that for any infinite set of unit vectors $U$ in $\ZR^2$ the maximal operator defined by $$ H_Uf(x)=\sup_{u\in U}\bigg|\pv\int_{-\infty}^\infty \frac{f(x-tu)}{t}dt\bigg|,\quad x\in \ZR^2, $$ is not bounded in $L^2(\ZR^2)$.
Let $Ω$ be any set of directions (unit vectors) on the plane. We study maximal operators defined by \md0 M_Ωf(x)=\sup_{δ>0, ω\in Ω} \frac{1}{2δ}\int_{-δ}^δ|f(x+tω)|dt. \emd for the generalized lacunary sets $Ω$ associated with an integer $μ>0$. It is proved the following sharp inequality: $$ \|M_Ωf(x)\|_2\lesssim \sqrtμ \|f\|_2. $$
We introduce a notion of (finite order) lacunarity in higher dimensions for which we can bound the associated directional maximal operators in $L^p(\mathbb{R}^n)$, with $p>1$. In particular, we are able to treat the classes previously considered by Nagel--Stein--Wainger, Sjögren--Sjölin and Carbery. Closely related to this, we find a characterisation of the sets of directions which give rise to bounded maximal operators. The bounds enable Lebesgue type differentiation of integrals in $L_{\text{loc}}^p(\mathbb{R}^n)$, replacing balls by tubes which point in these directions.
We introduce the notion of \textit{Perron capacity} of a set of slopes $Ω\subset \mathbb{R}$. Precisely, we prove that if the Perron capacity of $Ω$ is finite then the directional maximal operator associated $M_Ω$ is not bounded on $L^p(\mathbb{R}^2)$ for any $1 < p < \infty$. This allows us to prove that the set $$Ω_{ \boldsymbol{e}} =\left\{ \frac{\cos n}{n}: n\in \mathbb{N}^* \right\}$$ is not finitely lacunary which answers a question raised by A. Stokolos.
For any dimension $n \geq 2$, we consider the maximal directional Hilbert transform $\mathscr{H}_U$ on $\mathbb R^n$ associated with a direction set $U \subseteq \mathbb S^{n-1}$: \[ \mathscr{H}_Uf(x) := \frac{1}π \sup_{v \in U} \Bigl| \text{p.v.} \int f(x - tv) \, \frac{dt}{t}\Bigr|.\] The main result in this article asserts that for any exponent $p \in (1, \infty)$, there exists a positive constant $C_{p,n}$ such that for any finite direction set $U \subseteq \mathbb S^{n-1}$, \[||\mathscr{H}_U||_{p \rightarrow p} \geq C_{p,n} \sqrt{\log \#U}, \] where $\#U$ denotes the cardinality of $U$. A…
