Optimal lifting of Levi-degenerate hypersurfaces and applications to the Cauchy--Szeg\"o projection

TL;DR

This paper develops an optimized lifting method for Levi-degenerate hypersurfaces in complex space, enabling explicit Taylor expansions and Schatten class estimates for the Cauchy--Szeg"o projection's commutator, advancing analysis on complex hypersurfaces.

Contribution

It introduces a constructive, optimized lifting procedure for Levi-degenerate hypersurfaces to stratified Lie groups, providing explicit Taylor expansions and applications to operator estimates.

Findings

Explicit lifting to stratified Lie groups for Levi-degenerate hypersurfaces.

Derivation of an explicit Taylor expansion in the sub-Riemannian setting.

Establishment of Schatten class estimates for the Cauchy--Szeg"o projection commutator.

Abstract

We consider a family of Levi-degenerate finite type hypersurfaces in $\mathbb C^2$, where in general there is no group structure. We lift these domains to stratified Lie groups via a constructive proof, which optimizes the well-known lifting procedure to free Lie groups of general manifolds defined by Rothschild and Stein. This yields an explicit version of the Taylor expansion with respect to the horizontal vector fields induced by the sub-Riemannian structure on these hypersurfaces. Hence, as an application, we establish the Schatten class estimates for the commutator of the Cauchy--Szeg\"{o} projection with respect to a suitable quasi-metric defined on the hypersurface.