On the Sobolev-Poincare inequality of CR-manifolds

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The purpose is to study the CR-manifold with a contact structure conformal to the Heisenberg group. In our previous work WY, we have proved that if the Q'-curvature is nonnegative, and the integral of Q'-curvature is below the dimensional bound c₁', then we have the isoperimetric inequality. In this paper, we manage to drop the condition on the nonnegativity of the Q'-curvature. 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.