JournalsprimsVol. 22 , No. 1DOI 10.2977/prims/1195178372

Jensen Measures and Maximal Functions of Uniform Algebras

  • Cho-ichiro Matsuoka

    Doshisha University, Kyoto, Japan
Jensen Measures and Maximal Functions of Uniform Algebras cover

Abstract

Our purpose here is to seek on an arbitrary uniform algebra the class of representing measures which admit a certain maximal function for each log-envelope function defined on the maximal ideal space of the algebra. These maximal functions can be considered as a proper generalization of those that are associated with two-dimensional Brownian motion in the concrete algebras R(K). Most of the results already obtained from the probabilistic approach, e. g. Burkholder-Gundy-Silverstein inequalities, a weaker form of Fefferman's duality theorem etc., are valid for our maximal functions. The remarkable feature of our class of representing measures is that it is stable under the weak-star limit and the convex combination. In the concrete algebras R(K), if the harmonic measure and the Keldysh measure for a given point of K are different, then our class of representing measures that are supported on the topological boundary of K forms an infinite-dimensional weak-star compact convex set in the dual of C(K).