# Closed range estimates for $\bar\partial_b$ on CR manifolds of hypersurface type

## TL;DR

This paper establishes new, more general conditions called weak Y(q) for the closed range of the ar_b operator on CR manifolds of hypersurface type, with applications to estimates and regularity of harmonic forms.

## Contribution

It introduces the weak Y(q) condition, a broader and more verifiable criterion for closed range estimates of ar_b on CR manifolds of hypersurface type.

## Key findings

- Weak Y(q) condition is easier to verify than previous conditions.
- Established closed range estimates for ar_b in L^2 and Sobolev spaces.
- Applications include Szeg
51 projection estimates and regularity of harmonic forms.

## Abstract

The purpose of this paper is to establish sufficient conditions for closed range estimates on $(0,q)$-forms, for some fixed $q$, $1 \leq q \leq n-1$, 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 Szeg\"o 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.