Nicoara, Andreea C. Global regularity for \(\bar{\partial}_b\) on weakly pseudoconvex CR manifolds. (English) Zbl 1091.32017 Adv. Math. 199, No. 2, 356-447 (2006). In this paper the results of J. J. Kohn [Trans. Am. Math. Soc. 181, 273–292 (1973; Zbl 0276.35071) and Duke Math. J. 53, 525–545 (1986; Zbl 0609.32015)] are extended to compact, orientable, weakly pseudoconvex CR mainfolds embedded in \(\mathbb C^N\) and of codimension higher than one. The new concept of a CR plurisubharmonic weight function is used. With the help of this weight function and a carefully constructed covering of the CR manifold, a microlocal norm \(\langle | . | \rangle_t \) is defined which turns out to be equivalent to the \(L^2\) norm. The main results are the following: Let \(M^{2n-1}\) be a compact, orientable, weakly pseudoconvex manifold of dimension at least 5, embedded in \(\mathbb C^N \;\;(n\leq N),\) of codimension one or above, and endowed with the induced CR structure. Then the ranges of \(\overline \partial_b,\) its adjoint \(\overline \partial_{b,t}^*\) with respect to \(\langle | . | \rangle_t ,\) and the Kohn Laplacian defined as \(\square_{b,t}=\overline \partial_b \overline \partial_{b,t}^* + \overline \partial_{b,t}^* \overline \partial_b \) are closed in \(L^2\) and all \(H^s\) spaces for \(s>0\) if \(t\) is large enough, for \((p,q)\) forms such that \(0\leq q \leq n-3 \;, \;2\leq q \leq n-1,\) and \(1\leq q \leq n-2,\) respectively. If \(\alpha \) is a closed \((p,q)\) form, \( 1\leq q \leq n-2,\) then there exists a \((p,q-1)\) form \(\omega \) such that \(\overline \partial_b \omega = \alpha\) and if \(\alpha \in H^s\) for \(s\geq 0,\) then \(\omega \in H^s.\) If \(\alpha \in \mathcal C^{\infty } \) is a closed \((p,q)\) form, \( 1\leq q \leq n-2,\) then there exists a \((p,q-1)\) form \(\omega \in \mathcal C^{\infty }\) such that \(\overline \partial_b \omega = \alpha.\) Let \(H_0^{p,q}(M,\overline \partial_b ), \;H_s^{p,q}(M,\overline \partial_b ) , \) and \(H_{\infty}^{p,q}(M,\overline \partial_b )\) be the \(\overline \partial_b \) cohomology of \(M\) with respect to \(L^2, \;H^s,\) and \( \mathcal C^{\infty }\) coefficients, respectively. Then \(H_0^{p,q}(M,\overline \partial_b ), \;H_s^{p,q}(M,\overline \partial_b ),\) and \(H_{\infty }^{p,q}(M,\overline \partial_b )\) are finite for \(1\leq q \leq n-2\) and \(H_0^{p,q}(M,\overline \partial_b )\cong H_s^{p,q}(M,\overline \partial_b ) \cong H_{\infty }^{p,q}(M,\overline \partial_b )\) for all \(s>0.\) Reviewer: Fritz Haslinger (Wien) Cited in 2 ReviewsCited in 20 Documents MSC: 32W10 \(\overline\partial_b\) and \(\overline\partial_b\)-Neumann operators 32V15 CR manifolds as boundaries of domains 32T25 Finite-type domains Keywords:embedded CR manifold; closed range of \(\bar{\partial_b}\); microlocalization Citations:Zbl 0276.35071; Zbl 0609.32015 PDFBibTeX XMLCite \textit{A. C. Nicoara}, Adv. Math. 199, No. 2, 356--447 (2006; Zbl 1091.32017) Full Text: DOI References: [1] Boas, H. P.; Shaw, M. C., Sobolev estimates for the Lewy operator on weakly pseudoconvex boundaries, Math. 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