Kakeya Maximal Operator Research Articles

Norm Estimates for the Kakeya Maximal Function in High Dimensions with Respect to General Measures

Norm Estimates for the Kakeya Maximal Function in High Dimensions with Respect to General Measures

New Kakeya estimates using Gromov's algebraic lemma

This paper presents several new results related to the Kakeya problem. First, we establish a geometric inequality which says that collections of direction-separated tubes (thin neighborhoods of line segments that point in different directions) cannot cluster inside thin neighborhoods of low degree algebraic varieties. We use this geometric inequality to obtain a new family of multilinear Kakeya estimates for direction-separated tubes. Using the linear / multilinear theory of Bourgain and Guth, these multilinear Kakeya estimates are converted into Kakeya maximal function estimates. Specifically, we obtain a Kakeya maximal function estimate in Rn at dimension d(n)=(2−2)n+c(n) for some c(n)>0. Our bounds are new in all dimensions except n=2,3,4, and 6.

A Kakeya maximal function estimate in four dimensions using planebrushes

We obtain an improved Kakeya maximal function estimate in \mathbb R^4 using a new geometric argument called the planebrush. A planebrush is a higher dimensional analogue of Wolff’s hairbrush, which gives effective control on the size of Besicovitch sets when the lines through a typical point concentrate into a plane. When Besicovitch sets do not have this property, the existing trilinear estimates of Guth–Zahl can be used to bound the size of a Besicovitch set. In particular, we establish a maximal function estimate in \mathbb R^4 at dimension 3.059. As a consequence, every Besicovitch set in \mathbb R^4 must have Hausdorff dimension at least 3.059.

Finite field restriction estimates based on Kakeya maximal operator estimates

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 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.

On the restriction theorem for the paraboloid in ${\mathbb R}^4$

We prove that the recent breaking (Zahl, 2018) of the $3/2$ barrier in Wolff’s estimate on the Kakeya maximal operator in ${\mathbb R}^4$ leads to improving the $14/5$ threshold for the restriction problem for the paraboloid in ${\mathbb R}^4$. One o

A discretized Severi-type theorem with applications to harmonic analysis

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 $\delta^{-2-\varepsilon}$ direction-separated $\delta$-tubes can be contained in the $\delta$-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 Guth and the author, this result implies a Kakeya maximal function estimate at dimension $3+1/28$, which is an improvement over the previous bound of $3$ due to Wolff. As a consequence, we prove that every Besicovitch set in $\mathbb{R}^4$ must have Hausdorff dimension at least $3+1/28$. Recently, Demeter showed that any improvement over Wolff's bound for the Kakeya maximal function yields new bounds on the restriction problem for the paraboloid in $\mathbb{R}^4$.

Polynomial Wolff axioms and Kakeya-type estimates in R4

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 $\delta$-tubes satisfies the Wolff axioms if not too many tubes can be contained in the $\delta$-neighborhood of a plane. A collection of tubes satisfies the polynomial Wolff axioms if not too many tubes can be contained in the $\delta$-neighborhood of a low degree algebraic variety. First, we prove that if a set of $\delta^{-3}$ tubes in $\mathbb{R}^4$ satisfies the polynomial Wolff axioms, then the union of the tubes must have volume at least $\delta^{1-1/40}$. We also prove a more technical statement which is analogous to a maximal function estimate at dimension $3+1/40$. Second, we prove that if a collection of $\delta^{-3}$ tubes in $\mathbb{R}^4$ satisfies the polynomial Wolff axioms, and if most triples of intersecting tubes point in three linearly independent directions, then the union of the tubes must have volume at least $\delta^{3/4}$. Again, we also prove a slightly more technical statement which is analogous to a maximal function estimate at dimension $3+1/4$. We conjecture that every Kakeya set satisfies the polynomial Wolff axioms, but we are unable to prove this. If our conjecture is correct, it implies a Kakeya maximal function estimate at dimension $3+1/40$, and in particular this implies that every Kakeya set in $\mathbb{R}^4$ must have Hausdorff dimension at least $3+1/40$. This would be an improvement over the current best bound of 3, which was established by Wolff in 1995.

A note on the Kakeya maximal operator and radial weights on the plane

We obtain an estimate of the operator norm of the weighted Kakeya (Nikodým) maximal operator without dilation on $L^2(w)$. Here we assume that a radial weight $w$ satisfies the doubling and supremum condition. Recall that, in the definition of the Kakeya maximal operator, the rectangle in the supremum ranges over all rectangles in the plane pointed in all possible directions and having side lengths $a$ and $aN$ with $N$ fixed. We are interested in its eccentricity $N$ with $a$ fixed. We give an example of a non-constant weight showing that $\sqrt{\log N}$ cannot be removed.

Quasisymmetric Maps on Kakeya Sets

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 a.e. line segment inside $K$ has dimension at most $\min\{2n/(n + 2),7n/(4n + 3)\}$. The Kakeya maximal function conjecture implies that the bounds can be improved to $n$ and $1$, respectively.

Heisenberg Hausdorff Dimensionof Besicovitch Sets

Abstract We consider (bounded) Besicovitch sets in the Heisenberg group and prove that Lp estimates forthe Kakeya maximal function imply lower bounds for their Heisenberg Hausdorff dimension.

Singular integrals along 𝑁 directions in ℝ²

We prove optimal bounds in L 2 ( R 2 ) L^2(\mathbb {R}^2) for the maximal operator obtained by taking a singular integral along N N arbitrary directions in the plane. We also give a new proof for the optimal L 2 L^2 bound for the single scale Kakeya maximal function in the plane.

THE KAKEYA SET AND MAXIMAL CONJECTURES FOR ALGEBRAIC VARIETIES OVER FINITE FIELDS

Using the polynomial method of Dvir \cite{dvir}, we establish optimal estimates for Kakeya sets and Kakeya maximal functions associated to algebraic varieties $W$ over finite fields $F$. For instance, given an $n-1$-dimensional projective variety $W \subset \P^n(F)$, we establish the Kakeya maximal estimate $$ \| \sup_{\gamma \ni w} \sum_{v \in \gamma(F)} |f(v)| \|_{\ell^n(W)} \leq C_{n,W,d} |F|^{(n-1)/n} \|f\|_{\ell^n(F^n)}$$ for all functions $f: F^n \to \R$ and $d \geq 1$, where for each $w \in W$, the supremum is over all irreducible algebraic curves in $F^n$ of degree at most $d$ that pass through $w$ but do not lie in $W$, and with $C_{n,W,d}$ depending only on $n, d$ and the degree of $W$; the special case when $W$ is the hyperplane at infinity in particular establishes the Kakeya maximal function conjecture in finite fields, which in turn strengthens the results of Dvir.

The Universal maximal operator on special classes of functions

We prove pointwise inequalities for the maximal operator over all the directions in R n when acting on l q -radial functions and on product functions. From these inequalities we deduce boundedness results on L p for p > n; these can be applied to other operators, in particular to the Kakeya maximal operator.

The Fefferman-Stein type inequality for the Kakeya maximal operator in Wolff’s range

Let K δ , 0 < δ << 1, be the Kakeya (Nikodym) maximal operator defined as the supremum of averages over tubes of eccentricity δ. The (so-called) Fefferman-Stein type inequality: ∥K δ f∥ Lp(Rd(Rd,w) ≤ C(1/δ) d/p-1 (log(1/δ)) α ∥f∥ Lp(Rd,Kδw) is shown in the range 1 < p < (d + 2)/2, where C and a are some constants depending only on p and the dimension d and w is a weight. The result is a sharp bound up to log(1/δ)-factors.

New bounds for Kakeya problems

We establish new estimates on the Minkowski and Hausdorff dimensions of Kakeya sets and we obtain new bounds on the Kakeya maximal operator.

Norm Estimates for the Kakeya Maximal Function with Respect to General Measures

We generalize Bourgain's theorem on the Kakeya maximal function in the plane by proving norm estimates with respect to measures satisfying certain conditions. We use this to extend the classical result of Davies on the Hausdor dimension of Kakeya sets in the plane.

A Weighted Inequality for the Kakeya Maximal Operator with a Special Base

In this paper we shall give a weighted version of Igari's estimate on the Kakeya maximal operator with a special base.

An Estimate for the Kakeya Maximal Operator on Functions of Square Radial Type

The small Kakeya maximal operator, $M_{a,N}$, in $\mathbf{R}^d$ is defined by averages on cylinders with the width $a$ and the height $Na$. We show that the inequality $\lVert M_{a,N}f\rVert_d \leq C\log N\lVert f\rVert_d$ holds for the functions of square radialy type, where $C$ is a constant depending only on $d$.

The Kakeya maximal operator with a special base

LetMℛ f be the Kakeya maximal function in d-dimensional Euclidean space with some base ℛ, consisting of cylinders of eccentricity N. The inequality ∥Mℛf∥d⩽c(logN)e∥ is shown for a base ℛ satisfying a direction condition, where e and c are constants depending only on d.

An elementary proof of an estimate for the Kakeya maximal operator on functions of product type

An elementary proof of an estimate for the Kakeya maximal operator on functions of product type