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
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
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
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
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
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
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
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
...
1
2
...

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
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
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
Representations of Integers as Sums of Squares
The general problem of determining exact formulas for r(s;n) is classical in number theory. One may consult the popular book by E. Grosswald [G] for a thorough account (as of the early 1980’s) of theExpand
A new form of the circle method, and its application to quadratic forms.
If the coefficients r(n) satisfy suitable arithmetic conditions the behaviour of F (α) will be determined by an appropriate rational approximation a/q to α, with small values of q usually producingExpand
An endpoint estimate for the discrete spherical maximal function
We prove that the discrete spherical maximal function extends to a bounded operator from L d/(d-2),1 (Z d ) to L d/(d-2), ∞(Z d ) in dimensions d > 5. This is an endpoint estimate for a recentExpand
On the distribution of lattice points on spheres and level surfaces of polynomials
Abstract The irregularities of distribution of lattice points on spheres and on level surfaces of polynomials are measured in terms of the discrepancy with respect to caps. It is found that theExpand
Analytic methods for Diophantine equations and Diophantine inequalities, by Harold Davenport
Harold Davenport was one of the truly great mathematicians of the twentieth century. Based on lectures he gave at the University of Michigan in the early 1960s, this book is concerned with the use ofExpand
...
1
2
3
...