In our dissertation we deal with the space H(K) of harmonic functions on a compact space in classical and abstract potential theory. Initially, we prove several equivalent characteristics of this space in classical potential theory. The internal characterization, which describes H(K) as a subspace of those continuous functions on a compact space K which are finely harmonic on the fine interior of K, is then used as the definition of H(K) in abstract potential theory. Further we concentrate on the solution of the Dirichlet problem for open and compact sets mainly with regards to its relation to subclasses of Baire class one functions. The results, proved at first in classical potential theory, are later generalized to abstract potential theory. With a use of more elemen- tary tools we initially prove these results in harmonic spaces with the axiom of dominance and, subsequently, using stronger tools we generalize them to harmonic spaces with the axiom of polarity. We engage also in a more abstract problem of approximation by differen- ces of lower semicontinuous functions in a more general context of binormal topological spaces.

Title: Baire and Harmonic Functions Author: Petr Pošta Department: Department of Mathematical Analysis Supervisor: prof. RNDr. Jaroslav Lukeš, DrSc., Department of Mathematical Analysis Abstract: The present thesis consists of six research papers. The first four articles deal with topics related to potential theory, Baire-one functions and its important subclasses, in particular differences of semicontinuous functions. The first paper is devoted to the stability of the Dirichlet problem for which a new criterion in terms of Poisson equation is provided. The second paper improves the recent result obtained by Lukeš et al. It shows that the classical Dirichlet solution belongs to the B1/2 subclass of Baire-one functions. A generalization of this result to the abstract context of the Choquet theory on functions spaces is provided. Finally, an abstract Dirichlet problem for the boundary condition belonging to the class of differences of semincontinuous functions is discussed. The third paper concentrates on the Lusin-Menshov property and the approximation of Baire- one and finely continuous functions by differences of semicontinuous and finely continuous functions. It provides an exposition of topologies (various density topologies as well as the fine topologies in both linear and non-linear potential...