On the sobolev-poincaré inequality of cr-manifolds

The purpose is to study the CR-manifold with a contact structure conformal to the Heisenberg group. In our previous work [22], we have proved that if the Q'-curvature is nonnegative and the integral of Q'-curvature is below the dimensional bound c' 1, then we have the isoperimetric inequality. In this paper, we manage to deal with general contact structure conformal to the Heisenberg group, removing the condition that Q'-curvature is nonnegative. We prove that the volume form e4u is a strong A∞ weight. As a corollary, we prove the Sobolev-Poincaré inequality on a class of CR-manifolds with integrable Q'-curvature.