On the tangential gradient of the kernel of the double layer potential

In this paper we consider an elliptic operator with constant coefficients and we estimate the maximal function of the tangential gradient of the kernel of the double layer potential with respect to its first variable. As a consequence, we deduce the validity of a continuity property of the double layer potential in H\"{o}lder spaces on the boundary that extends previous results for the Laplace operator and for the Helmholtz operator.