Dimension-free estimates for discrete maximal functions related to normalized gaussians

Abstract

In this paper, we investigate dimension-free estimates for maximal operators of convolutions with discrete normalized Gaussians (related to the Theta function) in the context of maximal, jump and $r$-variational inequalities on $\ell^p(\mathbb{Z}^d)$ spaces. This is the first instance of a discrete operator in the literature where $\ell^p(\mathbb{Z}^d)$ bounds are provided for the entire range of $1 < p < \infty$. The methods of proof rely on developing robust Fourier methods, which are combined with the fractional derivative, a tool that has not been previously applied to studying similar questions in the discrete setting.