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

Abstract

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.