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arXiv · 1202.1511

ABC-type estimates via Garsia-type norms

Konstantin M. Dyakonov
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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.

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Konstantin M. Dyakonov. 2012-02-07. ABC-type estimates via Garsia-type norms. https://doi.org/10.1090/conm%2F561%2F11116

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