Gradient of solution of the Poisson equation in the unit ball and related operators

In this paper we determine the $L^1\to L^1$ and $L^{\mathrm{\infty }}\to L^{\mathrm{\infty }}$ norms of an integral operator $\mathcal{N}$ related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions.