A proof of Khavinson's conjecture in $\mathbf{R}^4$

Abstract

The paper deals with an extremal problem for bounded harmonic functions in the unit ball of $\mathbf{R}^4$. We solve the generalized Khavinson problem in $\mathbf{R}^4$. This precise problem was formulated by G. Kresin and V. Maz'ya for harmonic functions in the unit ball and in the half--space of $\mathbf{R}^n$. We find the optimal pointwise estimates for the norm of the gradient of bounded real--valued harmonic functions.