A Function Algebra Approach to a Theorem of Lindelof

We study this and associated theorems in the context of a logmodular function algebra and, in particular, in the Banach algebra, H, of bounded analytic functions on D. Those multiplicative linear functionals (homomorphisms) on if which are in the weak* closure of some Stolz angle at 1 are called Stolz homomorphisms. We call a homomorphism h a Lindelof homomorphism provided h(f) = 0 whenever ess lim/(e'°) = 0 as 0 -> 0 (or 0"). Theorem 1.1 states that every Stolz homomorphism is a Lindelof homomorphism. In §3 we prove that the Lindelof homomorphisms coincide with the weak* closure of the Stolz homomorphisms. We show that there is an open-closed subset, Jt, of the Silov boundary of #°° such that the Stolz homomorphisms are those homomorphisms in the fibre over 1 whose ArensSinger measure assigns Jt mass strictly between 0 and 1. By replacing Jt by an arbitrary open-closed subset of the Silov boundary we can extend the notions of both Stolz and Lindelof homomorphisms. We show that for any such Jt the Jt-Stolz homomorphisms form an open subset of the fibre over 1, and is a union of parts. We prove that the set of .///-Lindelof homomorphisms is the weak* closure of the Jt-Stolz homomorphisms. The main tool used in the related classical investigations is the concept of the harmonic measure of a measurable subset of the unit circle. In §2 we make the natural extension, employing Arens-Singer measures, to logmodular algebras and define the harmonic measure of a Borel subset of the Silov boundary. These generalized harmonic measures turn out to be harmonic on each non-trivial Gleason part and essentially provide an exact analogue of the classical concept. We then prove a generalized " w-constants theorem " from which we obtain a general Lindelof theorem which motivates the results of §3. The general Lindelof theorem is then utilized to prove a very general concrete Lindelof theorem due to Doob [6; Theorem 3]. Our terminology, unless otherwise stated is that of [8] and [9].