An $L^p$-comparison, $p\in (1,\infty)$, on the finite differences of a discrete harmonic function at the boundary of a discrete box

TL;DR

This paper establishes an $L^p$-comparison for finite differences of discrete harmonic functions at the boundary of a discrete box, extending continuous harmonic analysis results to a discrete setting with size-independent constants.

Contribution

It formulates and proves a discrete analogue of the continuous $L^p$-gradient comparison for harmonic functions, with bounds independent of the box size.

Findings

Discrete $L^p$-comparison for harmonic functions established

Constants do not depend on the size of the discrete box

Extends continuous harmonic analysis results to discrete lattice setting

Abstract

It is well-known that for a harmonic function $u$ defined on the unit ball of the $d$-dimensional Euclidean space, $d\geq 2$, the tangential and normal component of the gradient $\nabla u$ on the sphere are comparable by means of the $L^p$-norms, $p\in(1,\infty)$, up to multiplicative constants that depend only on $d,p$. This paper formulates and proves a discrete analogue of this result for discrete harmonic functions defined on a discrete box on the $d$-dimensional lattice with multiplicative constants that do not depend on the size of the box.