Tower multitype and global regularity of the $\bar\partial$-Neumann operator

TL;DR

This paper introduces a new approach to property (P_q) in the ar-Neumann problem, establishing regularity results for pseudoconvex hypersurfaces using novel invariants and formal techniques.

Contribution

It provides a self-contained proof of property (P_q) for finite D'Angelo q-type hypersurfaces, extending to more general classes with new invariants and formal methods.

Findings

Proved property (P_q) for pseudoconvex hypersurfaces of finite D'Angelo q-type.

Developed a new multitype invariant based on derivatives of the Levi form.

Introduced formal tools like supertangent vector fields and relative contact orders.

Abstract

A new approach is given to property $(P_q)$ defined by Catlin for $q=1$ in a global and by Sibony in a local context, subsequently extended by Fu-Straube for $q>1$. This property is known to imply compactness and global regularity in the $\bar\partial$-Neumann problem by a result of Kohn-Nirenberg, as well as condition $R$ by a result of Bell-Ligocka. In particular, we provide a self-contained proof of property $(P_q)$ for pseudoconvex hypersurfaces of finite D'Angelo $q$-type, the case originally studied by Catlin. Moreover, our proof covers more general classes of hypersurfaces inspired by a recent work of Huang-Yin. Proofs are broken down into isolated steps, some of which do not require pseudoconvexity. Our tools include: a new multitype invariant based on distinguished nested sequences of $(1,0)$ subbundles, defined in terms of derivatives of the Levi form; real and complex formal orbits; $k$-jets of functions relative to pairs of formal submanifolds; relative contact orders generalizing the usual contact orders; a new notion of supertangent vector fields having higher than expected relative contact orders; and a formal variant of a result by Diederich-Forn\ae ss arising as a key step in their proof of Kohn's ideal termination in the real-analytic case.