ABC-type estimates via Garsia-type norms

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.

9 pages

Cited by in corpus (1)

• On the singular factor of a linear combination of holomorphic functions