Zalcman's lemma, Pinchuk's rescaling method, and Catlin's estimates revisited

TL;DR

This paper introduces a new renormalization lemma for maps on the unit disc, unifies classical lemmas, and applies it to normality and metric estimates in complex analysis.

Contribution

It presents a generalized renormalization lemma that encompasses Zalcman and Miniowitz's results and applies it to Pinchuk's scaling method and Catlin's estimates.

Findings

Unified classical renormalization lemmas under a new framework

Established a general normality statement for Pinchuk's scaling method in C^2

Reproved Catlin's estimates for the Kobayashi metric in finite type domains

Abstract

We present a renormalization lemma for certain maps defined on the unit disc of C and taking values in some metric space. We show that the classical renormalization lemmas of Zalcman and Miniowitz can be deduced from our lemma. We also use it to establish a general normality statement for the Pinchuk's scaling method in C^2 and, incidentally, reprove the Catlin's estimates for the Kobayashi metric in finite type domains.