Multidimensional harmonic functions analogues of sharp real-part theorems in complex function theory

In the present paper, the sharp multidimensional analogues of Lindelöf inequality and similar estimates for analytic functions are considered. Using a sharp inequality for the gradient of a bounded or semibounded harmonic function in a ball, one arrives at improved estimates (compared with the known ones) for the gradient of harmonic functions in an arbitrary sub-domain of Rⁿ. A representation of the sharp constant in a pointwise estimate of the gradient of a harmonic function in a half-space is obtained under the assumption that function’s boundary values belong to L^p. This representation is realized in the three-dimensional case and the values of sharp constants are explicitly given for p = 1, 2, ∞.