The complex Green operator on CR-submanifolds of $\mathbb{C}^{n}$ of hypersurface type: compactness

TL;DR

This paper proves compactness estimates for the tangential Cauchy-Riemann operator on certain CR-submanifolds of complex space, extending known results from hypersurfaces to more general submanifolds using local equivalence techniques.

Contribution

It extends compactness estimates for the $ar{ ext{d}}_b$ operator from hypersurfaces to CR-submanifolds of hypersurface type, utilizing local CR-equivalence.

Findings

Establishes compactness estimates under property(P)

Shows local CR-equivalence to hypersurfaces implies similar estimates

Clarifies the relationship between potential theoretic conditions

Abstract

We establish compactness estimates for $\overline{\partial}_{b}$ on a compact pseudoconvex CR-submanifold of $\mathbb{C}^{n}$ of hypersurface type that satisfies property(P). When the submanifold is orientable, these estimates were proved by A.~Raich via microlocal methods. Our proof deduces the estimates from (a slight extension, when $q>1$, of) those known on hypersurfaces via the fact that locally, CR-submanifolds of hypersurface type are CR-equivalent to a hypersurface. The relationship between two potential theoretic conditions is also clarified.