Discrete Analogues in Harmonic Analysis: Directional Maximal Functions in $\mathbb{Z}^2$

TL;DR

This paper studies discrete directional maximal functions on , connecting harmonic analysis with geometric and number-theoretic methods, and explores an arithmetic Kakeya problem and polynomial orbit operators.

Contribution

It introduces new bounds for discrete directional maximal functions and links harmonic analysis with geometric measure theory and number theory.

Findings

Established bounds for directional maximal functions in 

Connected harmonic analysis problems with arithmetic Kakeya-type problems

Analyzed maximal operators along polynomial orbits in the discrete setting

Abstract

Let $V = \{ v_1,\dots,v_N\}$ be a collection of $N$ vectors that live near a discrete sphere. We consider discrete directional maximal functions on $\mathbb{Z}^2$ where the set of directions lies in $V$, given by \[ \sup_{v \in V, k \geq C \log N} \left| \sum_{n \in \mathbb{Z}} f(x-v\cdot n ) \cdot \phi_k(n) \right|, \ f:\mathbb{Z}^2 \to \mathbb{C}, \] where and $\phi_k(t) := 2^{-k} \phi(2^{-k} t)$ for some bump function $\phi$. Interestingly, the study of these operators leads one to consider an "arithmetic version" of a Kakeya-type problem in the plane, which we approach using a combination of geometric and number-theoretic methods. Motivated by the Furstenberg problem from geometric measure theory, we also consider a discrete directional maximal operator along polynomial orbits, \[ \sup_{v \in V} \left| \sum_{n \in \mathbb{Z}} f(x-v\cdot P(n) ) \cdot \phi_k(n) \right|, \ P \in \mathbb{Z}[-] \] for $k \geq C_d \log N$ sufficiently large.