Square function estimates for discrete Radon transforms

Abstract

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