Estimates for the complex Green operator: Symmetry, percolation, and interpolation

Abstract

Let $M$ be a pseudoconvex, oriented, bounded and closed CR submanifold of $\mathbb{C}^{n}$ of hypersurface type. We show that Sobolev estimates for the complex Green operator hold simultaneously for forms of symmetric bidegrees, that is, they hold for $(p,q)$--forms if and only if they hold for $(m-p,m-1-q)$--forms. Here $m$ equals the CR dimension of $M$ plus one. Symmetries of this type are known to hold for compactness estimates. We further show that with the usual microlocalization, compactness estimates for the positive part percolate up the complex, i.e. if they hold for $(p,q)$--forms, they also hold for $(p,q+1)$--forms. Similarly, compactness estimates for the negative part percolate down the complex. As a result, if the complex Green operator is compact on $(p,q_{1})$--forms and on $(p,q_{2})$--forms ($q_{1}\leq q_{2}$), then it is compact on $(p,q)$--forms for $q_{1}\leq q\leq q_{2}$. It is interesting to contrast this behavior of the complex Green operator with that of the $\overline{\partial}$--Neumann operator on a pseudoconvex domain.