  We show global and interior higher-order log-H\"older regularity estimates
for solutions of Dirichlet integral equations where the operator has a
nonintegrable kernel with a singularity at the origin that is weaker than that
of any fractional Laplacian. As a consequence, under mild regularity
assumptions on the right hand side, we show the existence of classical
solutions of Dirichlet problems involving the logarithmic Laplacian and the
logarithmic Schr\"odinger operator.
