It is well-known that for a harmonic function u defined on the unit ball of the d-dimensional Euclidean space, d ≥ 2, the tangential and normal component of the gradient ∇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.

中文翻译：

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一个L p的比较，p∈（1，∞）$ p \ in（1，\ infty）$，关于离散盒子边界上离散调和函数的有限差分

这是公知的，对于一个谐波函数ü在单位球定义d维欧几里得空间，d ≥2时，切向和梯度∇的法向分量ü通过的装置上的球是可比较的大号^p -范数\（p \ in（1，\ infty）\），直到仅取决于d，p的乘法常数。本文针对d维晶格上离散盒上定义的离散谐波函数公式化并证明了该结果的离散模拟，其中离散常数与盒的大小无关。