Jensen measures and maximal functions of uniform algebras
For any uniform algebra A and any point q of the maximal ideal space of A the author seeks a certain class \(T_ q\) of Jensen measures for q which admit a maximal function to any long-envelope function associated with A. The maximal functions can be considered as a proper generalization of Brownian maximal functions well-known in the classical Hardy spaces. Most of the results already obtained from the probabilistic approach, e.g. Burkholder-Gundy-Silverstein inequalities, Fefferman-Stein estimate, etc., are valid for our maximal functions. The remarkable feature of \(T_ q\) is that it is stable in the convex combination and weakstar limit. Various examples are discussed in the concrete algebra R(K). Further results are known: for any A and any q the class \(T_ q\) contains a measure which is supported on the Shilov boundary of A. Moreover, in the case \(A=R(K)\) the totality of Jensen measures for q is identical with \(T_ q\). The definition of \(T_ q\) is applicable under more general situation, e.g. any compact convex set in some locally convex topological vector space and the cone of continuous convex functions on that set. Then every probability measure on the convex set belongs to the class similar to \(T_ q\), where q is equal to the barycenter of the measure in question.
scientific article; zbMATH DE number 3985930