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

TL;DR

This work proves dimension-free, $\ell^p(\mathbb{Z}^d)$ bounds for discrete maximal, jump, variational, and oscillation inequalities associated with the discrete normalized Gaussian convolution $G_t$, valid for all $p\in(1,\infty)$ and independent of the dimension $d$. The authors combine Stein's complex interpolation with fractional derivatives and sharp Fourier-transform estimates of the kernel $g_t$—including a detailed analysis of small and large scales and a comparison to the discrete heat kernel—despite $G_t$ not forming a semigroup. They develop a robust framework for handling multiplier families via a dimension-free estimate for the difference operator, enabling variational and oscillation control in addition to maximal bounds. The results advance the understanding of dimension-free phenomena in the discrete setting and provide techniques that may inform dimension-free estimates for discrete ball and sphere averages, linking to Stein's longstanding question on the discrete case.

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.

Theorems & Definitions (45)

• Theorem 1.1
• Proposition 1.1
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Lemma 4.1
• proof
• Proposition 4.2
• Lemma 4.3