Jensen Measures and Maximal Functions of Uniform Algebras

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 $\mathbf{R}(K)$. 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 $\mathbf{R}(K)$, 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 $\mathbf{R}(K)$.