Let $M$ be a smooth formally integrable CR manifold of CR dimension $m$, and let $T^{0,1}M\subset \mathbb C TM$ denote its antiholomorphic CR bundle. Thus, for every [open set](/page/Open%20Set) $U\subset M$ and all local smooth sections $\bar L,\bar K\in C^\infty(U,T^{0,1}M)$, the complex Lie bracket $[\bar L,\bar K]$ is again a section of $T^{0,1}M$ over $U$.

For $0\le q\le m$, let $\Lambda_b^{0,q}M:=\Lambda^q((T^{0,1}M)^*)$ be the bundle of CR antiholomorphic $q$-forms, and set $\Lambda_b^{0,q}M:=0$ for $q>m$. Let

\begin{align*} \bar\partial_b:C^\infty(M,\Lambda_b^{0,q}M)\to C^\infty(M,\Lambda_b^{0,q+1}M) \end{align*}

be the tangential Cauchy-Riemann operator, defined by restricting the [exterior derivative](/theorems/1525) to antiholomorphic CR directions, with the outgoing map understood to be the zero map when the target bundle is zero. Then, for every integer $q$ with $0\le q\le m$,

\begin{align*} \bar\partial_b\circ\bar\partial_b=0:C^\infty(M,\Lambda_b^{0,q}M)\to C^\infty(M,\Lambda_b^{0,q+2}M). \end{align*}