Weighted inequalities and a.e. convergence for Poisson integrals in light-cones

We show that the Poisson maximal operator for the tube over the light-cone, P ^*, is bounded in the weighted space L ^p (w) if and only if the weight w(x) belongs to the Muckenhoupt class A _p . We also characterize with a geometric condition related to the intrinsic geometry of the cone the weights v(x) for which P ^* is bounded from L ^p (v) into L ^p (u), for some other weight u(x) > 0. Some applications to a.e. restricted convergence of Poisson integrals are given.