Bonsall, F. F. Domination of the supremum of a bounded harmonic function by its supremum over a countable subset. (English) Zbl 0658.31001 Proc. Edinb. Math. Soc., II. Ser. 30, 471-477 (1987). 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 \(H^{\infty}\). Cited in 2 ReviewsCited in 5 Documents MSC: 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions Keywords:domination; harmonic; nontangentially dense; harmonic measure; positive bounded Citations:Zbl 0096.051 × Cite Format Result Cite Review PDF Full Text: DOI 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 · doi:10.1093/qmath/37.2.129 [4] DOI: 10.2307/1993491 · Zbl 0096.05103 · doi:10.2307/1993491 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.