Let $M$ be a compact smooth formally integrable CR manifold of CR dimension $n$, equipped with a smooth positive density and Hermitian metrics on the bundles $\Lambda_b^{0,k}M$ for $0\le k\le n$. Set $H_{-1}=H_{n+1}=\{0\}$ and, for $0\le k\le n$, let
denote the maximal closed $L^2$ realization of the tangential Cauchy-Riemann operator on $(0,k)$-forms, and set $\bar\partial_{b,-1}=0$ and $\bar\partial_{b,n}=0$. Assume $\bar\partial_{b,k+1}\bar\partial_{b,k}=0$ for all admissible $k$. Let $\bar\partial_{b,k}^*:H_{k+1}\to H_k$ denote the Hilbert-space adjoint.
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Fix $q$ with $0\le q\le n$. Suppose that there exist constants $\varepsilon>0$ and $C>0$ such that every
Assume also the corresponding smooth subelliptic regularity in degree $q$: every $L^2$ harmonic $(0,q)$-form is smooth, and whenever $f\in C^\infty(M,\Lambda_b^{0,q}M)\cap\ker \bar\partial_{b,q}$ is orthogonal to the harmonic space, there exists $v\in C^\infty(M,\Lambda_b^{0,q-1}M)$ such that $\bar\partial_{b,q-1}v=f$, with the interpretation $C^\infty(M,\Lambda_b^{0,-1}M)=\{0\}$.
be the self-adjoint Kohn Laplacian on $H_q$, defined by its closed quadratic form on $\operatorname{Dom}(\bar\partial_{b,q})\cap\operatorname{Dom}(\bar\partial_{b,q-1}^*)$, and define
Then $\operatorname{Range}(\Box_{b,q})$ is closed in $H_q$, the space $\mathcal H_{b,L^2}^{0,q}(M)$ is finite-dimensional, and every element of $\mathcal H_{b,L^2}^{0,q}(M)$ is smooth.
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Moreover, the map $\mathcal H_{b,L^2}^{0,q}(M)\to H_{KR}^{0,q}(M)$ defined by $h\mapsto [h]$ from smooth harmonic forms to smooth Kohn-Rossi cohomology is an isomorphism, where