An interplay between real functions theory and potential theory

Jaroslav Lukeš

Charles University Prague

For an open bounded set $U$ of $\bold R ^m$, let the space $H(U)$ consist of all continuous functions on $\overline{U}$ which are harmonic on $U$. Given a continuous function $f$ on the boundary of $U$ denote by $f ^{CU}$ the Dirichlet solution of $f$. Further, let $\mathcal B _1^b1(H(U) )$ be the space of all bounded functions on $\overline{U}$ which are pointwise limits of functions from $H(U)$.

We show a close relation between some methods of real functions theory and potential theory. For example, we indicate proofs that the Dirichlet solution $f ^{CU}$ belongs to the space $\mathcal B _1^b(H(U))$. Moreover, we examine a question whether or not the space $\mathcal B _1^b(H(U))$ satisfies the ``barycentric formula'' or it is uniformly closed. Solving these problems we use in an essential way the fine topology methods and Choquet's theory of simplicial spaces.

The situation is quite different when replacing the function space $H(U)$ by the space of continuous affine functions on a compact convex set and $\mathcal B _1^b(H(U))$ by the space of Baire-one affine functions (positive theorems of Choquet and Mokobodzki).

The exposition will be quite elementary and all basic notions will be explained.