Global Regularity for the ∂ - b -Equation on C R Manifolds of Arbitrary Codimension

Abstract

Let $M$ be a ${\mathcal{C}}^{\mathrm{\infty }}$ compact $CR$ manifold of $CR$-codimension $\mathcal{l}\ge 1$ and $CR$-dimension $n-\mathcal{l}$ in a complex manifold $X$ of complex dimension $n\ge 3$. In this paper, assuming that $M$ satisfies condition $Y(s)$ for some $s$ with $1\le s\le n-\mathcal{l}-1$, we prove an $L^2$-existence theorem and global regularity for the solutions of the tangential Cauchy-Riemann equation for $(0,s)$-forms on $M$.