Malgrange's vanishing theorem for weakly pseudoconcave CR manifolds

The authors prove the following CR version of Malgrange's theorem: Assume $M$M is a smooth, non-compact, weakly pseudoconcave CR manifold of type $(n,k)$(n,k) of finite kind. Then the highest $\overline{\partial}_M$∂−M cohomology $H^{p,n}_{\overline{\partial}_M}(M)$Hp,n∂−M(M) vanishes for $0\le p\le n+k$0≤p≤n+k. This generalises a similar result for real analytic CR manifolds by the third author [in Hyperbolic problems and regularity questions, 137--150, Birkhäuser, Basel, 2007; MR2298789 (2008d:32034)]. Furthermore, they prove the following approximation theorem: If $M$M is as above and $U\subset\subset V \subset\subset M$U⊂⊂V⊂⊂M are two open sets such that $V\sbs U$V∖U has no compact connected component then for $0\le p\le n+k$0≤p≤n+k the restriction map $Z^{p,n-1}(\overline{V})\to Z^{p,n-1}(U)$Zp,n−1(V−)→Zp,n−1(U) has dense image, with respect to the $\scr C^\infty$C∞ topology on $U$U. The authors prove the following CR version of Malgrange's theorem: Assume $M$M is a smooth, non-compact, weakly pseudoconcave CR manifold of type $(n,k)$(n,k) of finite kind. Then the highest $\overline{\partial}_M$∂−M cohomology $H^{p,n}_{\overline{\partial}_M}(M)$Hp,n∂−M(M) vanishes for $0\le p\le n+k$0≤p≤n+k. This generalises a similar result for real analytic CR manifolds by the third author [in Hyperbolic problems and regularity questions, 137--150, Birkhäuser, Basel, 2007; MR2298789 (2008d:32034)]. Furthermore, they prove the following approximation theorem: If $M$M is as above and $U\subset\subset V \subset\subset M$U⊂⊂V⊂⊂M are two open sets such that $V\sbs U$V∖U has no compact connected component then for $0\le p\le n+k$0≤p≤n+k the restriction map $Z^{p,n-1}(\overline{V})\to Z^{p,n-1}(U)$Zp,n−1(V−)→Zp,n−1(U) has dense image, with respect to the $\scr C^\infty$C∞ topology on $U$U. The authors prove the following CR version of Malgrange's theorem: Assume $M$M is a smooth, non-compact, weakly pseudoconcave CR manifold of type $(n,k)$(n,k) of finite kind. Then the highest $\overline{\partial}_M$∂−M cohomology $H^{p,n}_{\overline{\partial}_M}(M)$Hp,n∂−M(M) vanishes for $0\le p\le n+k$0≤p≤n+k. This generalises a similar result for real analytic CR manifolds by the third author [in Hyperbolic problems and regularity questions, 137--150, Birkhäuser, Basel, 2007; MR2298789 (2008d:32034)]. Furthermore, they prove the following approximation theorem: If $M$M is as above and $U\subset\subset V \subset\subset M$U⊂⊂V⊂⊂M are two open sets such that $V\sbs U$V∖U has no compact connected component then for $0\le p\le n+k$0≤p≤n+k the restriction map $Z^{p,n-1}(\overline{V})\to Z^{p,n-1}(U)$Zp,n−1(V−)→Zp,n−1(U) has dense image, with respect to the $\scr C^\infty$C∞ topology on $U$U.