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

Abstract

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

arXiv comment27 pages. Final version incorporating the referee's suggestions. Accepted for publication in the Journal of Functional Analysis