Lower bound for the geometric type from a generalized estimate in the $\dib$-Neumann problem - a new approach by peak functions

TL;DR

This paper presents a new approach linking $f$-estimates in the $ar ext{ extbeta}$-Neumann problem to lower bounds on boundary geometric type, utilizing peak functions to simplify the proof.

Contribution

It introduces a simplified proof connecting $f$-estimates to geometric boundary bounds via peak functions, advancing understanding of boundary regularity.

Findings

$f$-estimates imply lower bounds on geometric type

Existence of peak functions is key to the proof

New approach simplifies previous methods

Abstract

We give a simple proof of the fact that an "$f$-estimate" for the $\bar\partial$-Neumann problem implies a lower bound on the geomatric type of the boundary along any complex one dimensional variety. The proof uses the existence of peak functions which is in turn a consequence of the $f$-estimate.