ABC-type estimates via Garsia-type norms

TL;DR

This paper extends the abc theorem from polynomials to analytic functions in the unit disk using Garsia-type norms, replacing zero counts with norms of associated Blaschke products in smoothness spaces.

Contribution

It introduces a new framework for abc-type estimates for analytic functions based on Garsia-type norms, especially in Lipschitz spaces, generalizing classical polynomial results.

Findings

Established abc-type estimates using Garsia-type norms.

Extended Mason–Stothers abc theorem to analytic functions.

Applied results to analytic Lipschitz spaces.

Abstract

We are concerned with extensions of the Mason--Stothers $abc$ theorem from polynomials to analytic functions on the unit disk $\mathbb D$. The new feature is that the number of zeros of a function $f$ in $\mathbb D$ gets replaced by the norm of the associated Blaschke product $B_f$ in a suitable smoothness space $X$. Such extensions are shown to exist, and the appropriate $abc$-type estimates are exhibited, provided that $X$ admits a "Garsia-type norm", i.e., a norm sharing certain properties with the classical Garsia norm on BMO. Special emphasis is placed on analytic Lipschitz spaces.