Square function estimates for discrete Radon transforms

TL;DR

This paper establishes $ ext{l}^p$ boundedness for discrete Radon transforms on integer lattices using square function estimates, extending techniques from continuous harmonic analysis to the discrete setting.

Contribution

It introduces a novel discrete Littlewood--Paley type approach to analyze Radon transforms, bridging continuous and discrete harmonic analysis methods.

Findings

Proves $ ext{l}^p$ boundedness for discrete Radon transforms

Develops square function estimates analogous to continuous Littlewood--Paley theory

Provides a powerful framework for discrete harmonic analysis

Abstract

We show $\ell^p\big(\mathbb Z^d\big)$ boundedness, for $p\in(1, \infty)$, of discrete singular integrals of Radon type with the aid of appropriate square function estimates, which can be thought as a discrete counterpart of the Littlewood--Paley theory. It is a very powerful approach which allows us to proceed as in the continuous case.