The discrete spherical averages over a family of sparse sequences

@article{Hughes2016TheDS,
title={The discrete spherical averages over a family of sparse sequences},
author={Kevin A. Hughes},
journal={arXiv: Classical Analysis and ODEs},
year={2016}
}
We initiate the study of the $\ell^p(\mathbb{Z}^d)$-boundedness of the arithmetic spherical maximal function over sparse sequences. We state a folklore conjecture for lacunary sequences, a key example of Zienkiewicz and prove new bounds for a family of sparse sequences that achieves the endpoint of the Magyar--Stein--Wainger theorem for the full discrete spherical maximal function. Perhaps our most interesting result is the boundedness of a discrete spherical maximal function in $\mathbb{Z}^4… Expand Restricted weak-type endpoint estimates for k-spherical maximal functions In this paper, we use the Approximation Formula for the Fourier transform of the solution set of lattice points on k-spheres and methods of Bourgain and Ionescu to refine the$$\ell ^p(\mathbbExpand Dimension-free estimates for the discrete spherical maximal functions • Mathematics • 2020 Abstract. We prove that the discrete spherical maximal functions (in the spirit of Magyar, Stein and Wainger) corresponding to the Euclidean spheres in Z with dyadic radii have l(Z) bounds for all pExpand Sparse bounds for the discrete spherical maximal functions • Mathematics • 2018 We give a short proof of some sparse bounds for the spherical maximal operator of Magyar, Stein and Wainger. This proof also includes certain endpoint estimates. The new method of proof is inspiredExpand$\ell ^p$-improving for discrete spherical averages In this paper, we prove$\ell^p$-improving estimates for the discrete spherical averages and some of their generalizations. At first glance this problem appears trivial, but upon further examinationExpand Bounds for Lacunary maximal functions given by Birch–Magyar averages • Mathematics • 2019 We obtain positive and negative results concerning lacunary discrete maximal operators defined by dilations of sufficiently nonsingular hypersurfaces arising from Diophantine equations in manyExpand Lacunary Discrete Spherical Maximal Functions • Mathematics • 2018 We prove new$\ell ^{p} (\mathbb Z ^{d})$bounds for discrete spherical averages in dimensions$ d \geq 5$. We focus on the case of lacunary radii, first for general lacunary radii, and then forExpand Lattice points problem, equidistribution and ergodic theorems for certain arithmetic spheres • Mathematics • 2021 We establish an asymptotic formula for the number of lattice points in the sets Sh1,h2,h3(λ) := {x ∈ Z 3 + : ⌊h1(x1)⌋+ ⌊h2(x2)⌋+ ⌊h3(x3)⌋ = λ} with λ ∈ Z+; where functions h1, h2, h3 are constantExpand ℓp-Improving Inequalities for Discrete Spherical Averages • Physics • 2020 Let λ2 ∈ ℕ, and in dimensions d ≥ 5, let Aλf(x) denote the average of f: ℤd → ℝ over the lattice points on the sphere of radius λ centered at x. We prove lp improving properties of Aλ:Expand$ \ell ^{p}$-improving inequalities for Discrete Spherical Averages • Mathematics • 2018 Let$ \lambda ^2 \in \mathbb N $, and in dimensions$ d\geq 5$, let$ A_{\lambda } f (x)$denote the average of$ f \;:\; \mathbb Z ^{d} \to \mathbb R $over the lattice points on the sphere ofExpand References SHOWING 1-10 OF 24 REFERENCES$L^p$-bounds for spherical maximal operators on$\mathbb Z^n$We prove analogue statements of the spherical maximal theorem of E. M. Stein, for the lattice points Zn. We decompose the discrete spherical measures as an integral of Gaussian kernels st,e(x) =Expand Problems and Results related to Waring's problem: Maximal functions and ergodic averages We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of \cite{Magyar_dyadic}, \cite{Magyar_ergodic} and \cite{MSW}. WeExpand Maximal functions and ergodic averages related to Waring’s problem We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of [13], [14] and [16]. We combine more precise knowledge ofExpand ENDPOINT MAPPING PROPERTIES OF SPHERICAL MAXIMAL OPERATORS • Mathematics • Journal of the Institute of Mathematics of Jussieu • 2003 For a function$f\in L^p(\mathbb{R}d)$,$d\ge 2$, let$A_tf(x)$be the mean of$f$over the sphere of radius$t$centred at$x$. Given a set$E\subset(0,\infty)\$ of dilations we prove variousExpand
Discrete analogues in harmonic analysis: Spherical averages
• Mathematics
• 2002
In this paper we prove an analogue in the discrete setting of d ,o fthe spherical maximal theorem for d . The methods used are two-fold: the application of certain “sampling” techniques, and ideasExpand
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