An L^p-comparison, p∈ (1,∞), on the finite differences of a discrete harmonic function at the boundary of a discrete box
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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∈(1,∞), 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.
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