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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:
[1] Dunford, Linear Operators Part I (1958) · Zbl 0088.32102
[2] Garnett, Bounded Analytic Functions (1981) · Zbl 0469.30024
[3] DOI: 10.1093/qmath/37.2.129 · Zbl 0594.46019
[4] DOI: 10.2307/1993491 · Zbl 0096.05103
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