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Domination of the supremum of a bounded harmonic function by its supremum over a countable subset. (English) Zbl 0658.31001
For what sequences $$\{a_ n\}$$ of points of the open unit disc D does there exist a constant $$\kappa$$ such that $(1)\quad \sup_{z\in D}| f(z)| \leq \kappa \sup_{n\in {\mathbb{N}}}| f(a_ n)|$ for all bounded harmonic functions f on D?
L. Brown, A. Shields and K. Zeller [Trans. Am. Math. Soc. 96, 162-183 (1960; Zbl 0096.051)] have proved the closely related result that $$(3)\quad \sup_{z\in D}| f(z)| =\sup_{n\in {\mathbb{N}}}| f(a_ n)|$$ for all $$f\in H^{\infty}$$ (the space of bounded analytic functions on D) if and only if $$\{a_ n\}$$ is nontangentially dense for $$\partial D$$, that is if and only if almost every point of $$\partial D$$ is the nontangential limit of some subsequence of $$\{a_ n\}$$. Our main result, Theorem 2, is a list of equivalent conditions on the sequence $$\{a_ n\}$$ which includes conditions (1) and (3).
In Theorem 3, we establish an elementary property of the harmonic measure $$\chi_ F(z)$$ of a Lebesgue measurable subset F of $${\mathbb{R}}$$; namely, $$\chi_ F(z)$$ is arbitrarily small outside the union of certain triangular domains associated with the points of F. This shows that if the inequality (1) holds for all positive bounded harmonic functions, then $$\{a_ n\}$$ is nontangentially dense.
Theorem 2 describes the sequences $$\{a_ n\}$$ for which the bounded linear mapping T of $$\ell^ 1$$ ito $$L^ 1$$ given by $$T\{\lambda_ n\}=\sum^{\infty}_{n=1}\lambda_ np_{a_ n}$$ is surjective. It is an immediate consequence that T is never bijective. When is it injective? This question remains unanswered, but Theorem 6 shows that T has zero kernel and closed range if and only if $$\{a_ n\}$$ is an interpolating sequence for $$H^{\infty}$$.

##### MSC:
 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
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##### References:
  Dunford, Linear Operators Part I (1958) · Zbl 0088.32102  Garnett, Bounded Analytic Functions (1981) · Zbl 0469.30024  DOI: 10.1093/qmath/37.2.129 · Zbl 0594.46019  DOI: 10.2307/1993491 · Zbl 0096.05103
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