Sobolev inequalities for $(0,q)$ forms on CR manifolds of finite type

TL;DR

This paper establishes sharp Sobolev inequalities for the $ar{ ext{d}}_b$ complex on certain compact CR manifolds, extending $L^1$ estimates for $(0,q)$ forms under specific geometric conditions.

Contribution

It introduces new $L^1$ Sobolev inequalities for the $ar{ ext{d}}_b$ complex on CR manifolds of finite type, including a novel $L^1$ duality inequality for vector fields satisfying Hormander's condition.

Findings

Proved sharp $L^1$ Sobolev inequalities for $(0,q)$ forms when $q eq 1, n-1$.

Established analogous inequalities under the $Y(q)$ condition.

Developed a new $L^1$ duality inequality for vector fields satisfying Hormander's condition.

Abstract

Let $M^{2n+1}$ ($n \geq 2$) be a compact pseudoconvex CR manifold of finite commutator type whose $\dbarb$ has closed range in $L^2$ and whose Levi form has comparable eigenvalues. We prove a sharp $L^1$ Sobolev inequality for the $\dbarb$ complex for $(0,q)$ forms when $q \ne 1$ nor $n-1$. We also prove an analogous $L^1$ inequality when $M$ satisfies condition $Y(q)$. The main technical ingredient is a new kind of $L^1$ duality inequality for vector fields that satisfy Hormander's condition.