Necessary conditions for local subellipticity of \(\square _ b\) on CR manifolds (Q1099855)

The author studies local a priori regularity estimates for the Laplacian \(\square _ b\) associated to the differential complex of tangential q- forms on a weakly pseudoconvex CR manifold of hypersurface type. The main theorem asserts that a local a priori estimate holds in the Sobolev norm \(\| \| _{\epsilon}\) only if the CR-manifold has a certain geometrical property. The geometrical property is that no almost-CR map imbed the CR manifold into \({\mathbb{C}}^ n \)as a hypersurface that admits osculation of order exceeding 1/\(\epsilon\) by a sequence of q-dimensional patches of complex submanifolds. The hypotheses of the main theorem include the condition that the basic global L 2 existence theory has been established for \(\square _ b\) on the CR manifold.