Regularity results for $\bar\partial_b$ on CR-manifolds of hypersurface type

TL;DR

This paper establishes regularity and solvability results for the $ard_b$ operator on a new class of CR manifolds called weak $Y(q)$, using microlocal analysis and weighted norms.

Contribution

It introduces the weak $Y(q)$ condition for CR manifolds and proves the closed range property and continuity of the Green operator in $L^2$ and Sobolev spaces.

Findings

$ard_b$ has closed range on $L^2$ for weak $Y(q)$ CR manifolds.

The complex Green operator is continuous on $L^2$ and Sobolev spaces.

The $ard_b$ equation can be solved in $C^inity$ on these manifolds.

Abstract

We introduce a class of embedded CR manifolds satisfying a geometric condition that we call weak $Y(q)$. For such manifolds, we show that dbar-b has closed range on $L^2$ and that the complex Green operator is continuous on $L^2$. Our methods involves building a weighted norm from a microlocal decomposition. We also prove that at any Sobolev level there is a weight such that the complex Green operator inverting the weighted Kohn Laplacian is continuous. Thus, we can solve the dbar-b equation in $C^\infty$.