# Closed Range Estimates for \(\bar{\partial }_b\) on CR Manifolds of Hypersurface Type

## Abstract

The purpose of this paper is to establish sufficient conditions for closed range estimates on (0, *q*)-forms, for some *fixed**q*, \(1 \le q \le n-1\), for \(\bar{\partial }_b\) in both \(L^2\) and \(L^2\)-Sobolev spaces in embedded, not necessarily pseudoconvex CR manifolds of hypersurface type. The condition, named weak *Y*(*q*), is both more general than previously established sufficient conditions and easier to check. Applications of our estimates include estimates for the Szegö projection as well as an argument that the harmonic forms have the same regularity as the complex Green operator. We use a microlocal argument and carefully construct a norm that is well suited for a microlocal decomposition of form. We do not require that the CR manifold is the boundary of a domain. Finally, we provide an example that demonstrates that weak *Y*(*q*) is an easier condition to verify than earlier, less general conditions.

## Keywords

Weak*Z*(

*q*) Weak

*Y*(

*q*) Tangential Cauchy–Riemann operator \(\bar{\partial }_b\) Closed range Microlocal analysis

## Mathematics Subject Classification

Primary 32W10 Secondary 32F17 32V20 35A27 35N15## Notes

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