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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.
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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 Rogers in dimension $n=3$. Our result is the first sharp estimate for maximal directional singular integrals in dimensions greater than 2. The proof relies on a representation of the maximal directional Hilbert transform in terms of a model maximal operator associated to compositions of two-dimensional angular multipliers, as well as on the usage of weighted norm inequalities, and their extrapolation, in the directional setting.
We study maximal operators related to bases on the infinite-dimensional torus $mathbb{T}^omega$. {For the normalized Haar measure $dx$ on $mathbb{T}^omega$ it is known that $M^{mathcal{R}_0}$, the maximal operator associated with the dyadic basis $mathcal{R}_0$, is of weak type $(1,1)$, but $M^{mathcal{R}}$, the operator associated with the natural general basis $mathcal{R}$, is not. We extend the latter result to all $q in [1,infty)$. Then we find a wide class of intermediate bases $mathcal{R}_0 subset mathcal{R} subset mathcal{R}$, for which maximal functions have controlled, but sometimes very peculiar behavior.} Precisely, for given $q_0 in [1, infty)$ we construct $mathcal{R}$ such that $M^{mathcal{R}}$ is of restricted weak type $(q,q)$ if and only if $q$ belongs to a predetermined range of the form $(q_0, infty]$ or $[q_0, infty]$. Finally, we study the weighted setting, considering the Muckenhoupt $A_p^mathcal{R}(mathbb{T}^omega)$ and reverse Holder $mathrm{RH}_r^mathcal{R}(mathbb{T}^omega)$ classes of weights associated with $mathcal{R}$. For each $p in (1, infty)$ and each $w in A_p^mathcal{R}(mathbb{T}^omega)$ we obtain that $M^{mathcal{R}}$ is not bounded on $L^q(w)$ in the whole range $q in [1,infty)$. Since we are able to show that [ bigcup_{p in (1, infty)}A_p^mathcal{R}(mathbb{T}^omega) = bigcup_{r in (1, infty)} mathrm{RH}_r^mathcal{R}(mathbb{T}^omega), ] the unboundedness result applies also to all reverse Holder weights.
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.
In this paper, we determine the $L^p(mathbb{R})times L^q(mathbb{R})rightarrow L^r(mathbb{R})$ boundedness of the bilinear Hilbert transform $H_{gamma}(f,g)$ along a convex curve $gamma$ $$H_{gamma}(f,g)(x):=mathrm{p.,v.}int_{-infty}^{infty}f(x-t)g(x-gamma(t)) ,frac{textrm{d}t}{t},$$ where $p$, $q$, and $r$ satisfy $frac{1}{p}+frac{1}{q}=frac{1}{r}$, and $r>frac{1}{2}$, $p>1$, and $q>1$. Moreover, the same $L^p(mathbb{R})times L^q(mathbb{R})rightarrow L^r(mathbb{R})$ boundedness property holds for the corresponding (sub)bilinear maximal function $M_{gamma}(f,g)$ along a convex curve $gamma$ $$M_{gamma}(f,g)(x):=sup_{varepsilon>0}frac{1}{2varepsilon}int_{-varepsilon}^{varepsilon}|f(x-t)g(x-gamma(t))| ,textrm{d}t.$$
We study both averaging and maximal averaging problems for Product $j$-varieties defined by $Pi_j={xin mathbb F_q^d: prod_{k=1}^d x_k=j}$ for $jin mathbb F_q^*,$ where $mathbb F_q^d$ denotes a $d$-dimensional vector space over the finite field $mathbb F_q$ with $q$ elements. We prove the sharp $L^pto L^r$ boundedness of averaging operators associated to Product $j$-varieties. We also obtain the optimal $L^p$ estimate for a maximal averaging operator related to a family of Product $j$-varieties ${Pi_j}_{jin mathbb F_q^*}.$
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