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Register a new userLogarithmic dimension bounds for the maximal function along a polynomial curve
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Authors
Ioannis Parissis
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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 d ||f||_2, where c>0 is an absolute constant. The proof depends on the explicit construction of a parabolic semi-group of operators which is a mixture of stable semi-groups.
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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 has not been used in the proof of sparse bounds before. The Hardy-Littlewood Circle method is used to decompose the multiplier into major and minor arc components. The efficiency arises as one only needs a single estimate on each element of the decomposition.
We establish the sharp growth order, up to epsilon losses, of the $L^2$-norm of the maximal directional averaging operator along a finite subset $V$ of a polynomial variety of arbitrary dimension $m$, in terms of cardinality. This is an extension of the works by Cordoba, for one-dimensional manifolds, Katz for the circle in two dimensions, and Demeter for the 2-sphere. For the case of directions on the two-dimensional sphere we improve by a factor of $sqrt{log N}$ on the best known bound, due to Demeter, and we obtain a sharp estimate for our model operator. Our results imply new $L^2$-estimates for Kakeya-type maximal functions with tubes pointing along polynomial directions. Our proof technique is novel and in particular incorporates an iterated scheme of polynomial partitioning on varieties adapted to directional operators, in the vein of Guth, Guth-Katz, and Zahl.
Let $mathsf M_{mathsf S}$ denote the strong maximal operator on $mathbb R^n$ and let $w$ be a non-negative, locally integrable function. For $alphain(0,1)$ we define the weighted sharp Tauberian constant $mathsf C_{mathsf S}$ associated with $mathsf M_{mathsf S}$ by $$ mathsf C_{mathsf S} (alpha):= sup_{substack {Esubset mathbb R^n 0<w(E)<+infty}}frac{1}{w(E)}w({xinmathbb R^n:, mathsf M_{mathsf S}(mathbf{1}_E)(x)>alpha}). $$ We show that $lim_{alphato 1^-} mathsf C_{mathsf S} (alpha)=1$ if and only if $win A_infty ^*$, that is if and only if $w$ is a strong Muckenhoupt weight. This is quantified by the estimate $mathsf C_{mathsf S}(alpha)-1lesssim_{n} (1-alpha)^{(cn [w]_{A_infty ^*})^{-1}}$ as $alphato 1^-$, where $c>0$ is a numerical constant; this estimate is sharp in the sense that the exponent $1/(cn[w]_{A_infty ^*})$ can not be improved in terms of $[w]_{A_infty ^*}$. As corollaries, we obtain a sharp reverse Holder inequality for strong Muckenhoupt weights in $mathbb R^n$ as well as a quantitative imbedding of $A_infty^*$ into $A_{p}^*$. We also consider the strong maximal operator on $mathbb R^n$ associated with the weight $w$ and denoted by $mathsf M_{mathsf S} ^w$. In this case the corresponding sharp Tauberian constant $mathsf C_{mathsf S} ^w$ is defined by $$ mathsf C_{mathsf S} ^w alpha) := sup_{substack {Esubset mathbb R^n 0<w(E)<+infty}}frac{1}{w(E)}w({xinmathbb R^n:, mathsf M_{mathsf S} ^w (mathbf{1}_E)(x)>alpha}).$$ We show that there exists some constant $c_{w,n}>0$ depending only on $w$ and the dimension $n$ such that $mathsf C_{mathsf S} ^w (alpha)-1 lesssim_{w,n} (1-alpha)^{c_{w,n}}$ as $alphato 1^-$ whenever $win A_infty ^*$ is a strong Muckenhoupt weight.
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.
Consider averages along the prime integers $ mathbb P $ given by begin{equation*} mathcal{A}_N f (x) = N ^{-1} sum_{ p in mathbb P ;:; pleq N} (log p) f (x-p). end{equation*} These averages satisfy a uniform scale-free $ ell ^{p}$-improving estimate. For all $ 1< p < 2$, there is a constant $ C_p$ so that for all integer $ N$ and functions $ f$ supported on $ [0,N]$, there holds begin{equation*} N ^{-1/p }lVert mathcal{A}_N frVert_{ell^{p}} leq C_p N ^{- 1/p} lVert frVert_{ell^p}. end{equation*} The maximal function $ mathcal{A}^{ast} f =sup_{N} lvert mathcal{A}_N f rvert$ satisfies $ (p,p)$ sparse bounds for all $ 1< p < 2$. The latter are the natural variants of the scale-free bounds. As a corollary, $ mathcal{A}^{ast} $ is bounded on $ ell ^{p} (w)$, for all weights $ w$ in the Muckenhoupt $A_p$ class. No prior weighted inequalities for $ mathcal{A}^{ast} $ were known.
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