Subellipticity of the $\bar \partial$-Neumann problem on nonpseudoconvex domains

Abstract:

Following the work of Kohn, we give a sufficient condition for subellipticity of the $\overline \partial$-Neumann problem for not necessarily pseudoconvex domains. We define a sequence of ideals of germs and show that if $1$ is in any of them, then there is a subelliptic estimate. In particular, we prove subellipticity under some specific conditions for $n - 1$ forms and for the case when the Levi-form is diagonalizable. For the necessary conditions, we use another method to prove that there is no subelliptic estimate for $q$ forms if the Levi-form has $n - q - 1$ positive eigenvalues and $q$ negative eigenvalues. This was proved by Derridj. Using similar techniques, we prove a necessary condition for subellipticity for some special domains. Finally, we give a remark to Catlin’s theorem concerning the hypoellipticity of the $\overline \partial$-Neumann problem in the case of nonpseudoconvex domains.