ﻻ يوجد ملخص باللغة العربية
We adapt Guths 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 $nge 7$ and improved bounds for the Kakeya set conjecture for an infinite sequence of dimensions.
We prove sparse bounds for the spherical maximal operator of Magyar, Stein and Wainger. The bounds are conjecturally sharp, and contain an endpoint estimate. The new method of proof is inspired by ones by Bourgain and Ionescu, is very efficient, and
The upper and lower Assouad dimensions of a metric space are local variants of the box dimensions of the space and provide quantitative information about the `thickest and `thinnest parts of the set. Less extre
We establish the sharp growth rate, in terms of cardinality, of the $L^p$ norms of the maximal Hilbert transform $H_Omega$ along finite subsets of a finite order lacunary set of directions $Omega subset mathbb R^3$, answering a question of Parcet and
Let M denote the maximal function along the polynomial curve p(t)=(t,t^2,...,t^d) in R^d: M(f)=sup_{r>0} (1/2r) int_{|t|<r} |f(x-p(t))| dt. We show that the L^2-norm of this operator grows at most logarithmically with the parameter d: ||M||_2 < c log
The purpose of this paper is to extend the result of arXiv:1810.00823 to mixed Holder functions on $[0,1]^d$ for all $d ge 1$. In particular, we prove that by sampling an $alpha$-mixed Holder function $f : [0,1]^d rightarrow mathbb{R}$ at $sim frac{1