ABC-type estimates via Garsia-type norms

Details ABC-type estimates via Garsia-type norms ABC-type estimates via Garsia-type norms

2012-02-07

arxiv.org/abs/1202.1511v1

In: Topics in Complex Analysis and Operator Theory. Contemporary Mathematics 561 (2012), 229-237

Mathematics

Complex Variables

9 pages

Scientific paper

10.1090/conm/561/11116

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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