On the Sobolev-Poincare inequality of CR-manifolds

TL;DR

This paper establishes a Sobolev-Poincaré inequality on certain CR-manifolds by showing the volume form is a strong $A_ abla$ weight, removing previous curvature sign restrictions.

Contribution

It proves the Sobolev-Poincaré inequality for CR-manifolds with integrable $Q'$-curvature without requiring nonnegativity, extending previous results.

Findings

Volume form $e^{4u}$ is a strong $A_ abla$ weight

Sobolev-Poincaré inequality holds under relaxed curvature conditions

Removed the nonnegativity condition on $Q'$-curvature

Abstract

The purpose is to study the CR-manifold with a contact structure conformal to the Heisenberg group. In our previous work \cite{WY}, 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 drop the condition on the nonnegativity of the $Q'$-curvature. We prove that the volume form $e^{4u}$ is a strong $A_\infty$ weight. As a corollary, we prove the Sobolev-Poincar\'e inequality on a class of CR-manifolds with integrable $Q'$-curvature.