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

It is well-known that for a harmonic function $u$ defined on the unit ball of the $d$-dimensional Euclidean space, $d\ge 2$, the tangential and normal component of the gradient $\mathrm{\nabla }u$ on the sphere are comparable by means of the $L^p$-norms, $p\in (1,\mathrm{\infty })$, 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.