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

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.