Global regularity of the tangential Cauchy-Riemann operator on weakly pseudoconvex CR manifolds.

摘要

Let M be a 2n - 1 dimensional CR manifold that is embedded in CN for n ≤ N, namely it is of codimension one or higher. In order to handle such a manifold of codimension higher than one, we introduce a new type of plurisubharmonicity, which we call CR plurisubharmonicity. If M is compact, orientable, weakly pseudoconvex, and of dimension at least 5, we are then able to prove using microlocalization with a CR plurisubharmonic function as a weight some estimates that imply the range of the tangential Cauchy-Riemann operator 6&d1;b is closed in L2 and any Sobolev space Hs with s > 0 for (p, q) forms with 0 ≤ q ≤ n - 3.; The 6&d1;b problem can thus be globally solved for the class of CR manifolds described above, and the solution turns out to be as regular as the datum. Moreover, the middle 6&d1;b cohomology groups of M with respect to L 2, Hs, and C infinity coefficients, Hp,q0M,6&d1; b , Hp,qsM,6&d1; b , and Hp,qinfinityM,6 &d1;b respectively, are finite and isomorphic to each other. In other words, even though for certain weakly pseudoconvex CR manifolds such as worm domains the canonical solution might not be smooth, it is still possible to give an indirect characterization of Hp,qinfinityM,6 &d1;b for 1 ≤ q ≤ n - 2.