Local regularity of the Green operator in a CR manifold of general "type"

Abstract

It is here proved that if a pseudoconvex CR manifold $M$ of hypersurface type has a certain "type", that we quantify by a vanishing rate $F$ at a submanifold of CR dimension $0$, then $\Box_b$ "gains $f^2$ derivatives" where $f$ is defined by inversion of $F$. Indeed the estimate is more accurate and it involves the Levi form of $M$ and of additional weights, instead of $\Box_b$. Next a general tangential estimate, "twisted" by a pseudodifferential operator $\Psi$ is established. The combination of the two yields a general "$f$-estimate" twisted by $\Psi$. We apply the twisted estimate for $\Psi$ which is the composition of a cut-off $\eta$ with a differentiation of order $s$ such as $R^s$ of Section 4. Under the assumption that $[\partial_b,\eta]$ and $[\partial_b,[\bar\partial_b,\eta]]$ are superlogarithmic multipliers in a sense inspired to Kohn, we get the local regularity of the Green operator $G=\Box_b^{-1}$. In particular, if $M$ has "infraexponential type" along $S\setminus\Gamma$ where $S$ is a manifold of CR dimension $0$ and $\Gamma$ a curve transversal to $T^{\mathbb C} M$, then we have local regularity of $G$. This gives an immediate proof of former work by Baracco, Khanh, Zampieri and by Kohn. The conclusion extends to "block decomposed" domains for whose blocks the above hypotheses hold separately. In the application of Section 4, $\Psi$ is composed by a cut off $\eta$ and a differentiation of order $s$ such as $\Lambda^s$ or $R^s$ and $M$ is a decoupled hypersurface which has infraexponential type along the coordinate lines $\mathbb R_{x_j}\setminus\{0\}$ and whose equations have differentials which are superlogarithmic multipliers in the sense of Kohn. In this situation, $\Box_b$ is locally hypoelliptic.