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

Let $M^{2n+1}$ ($n\ge 2$) be a compact pseudoconvex CR manifold of finite commutator type whose $\text{dbarb}$ has closed range in $L^2$ and whose Levi form has comparable eigenvalues. We prove a sharp $L^1$ Sobolev inequality for the $\text{dbarb}$ complex for $(0,q)$ forms when $q\mathrm{\ne }1$ 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.