Jürgen Bliedtner

The Best Approach for Boundary Limits

In [5], J. L. Doob showed that for positive harmonic functions on the unit disk, there is no coarsest approach neighborhood system, i.e. filter, for which a Fatou boundary limit theorem holds when the filter is copied by rotation at all points of the unit circle. In [4]the authors used the principal result from [3] to show that there is a coarsest filter when the problem is suitably normalized. The normalization assigns to each positive harmonic function a zero set; this is a set of boundary points at which the function must vanish. Known limits such as those provided by the Lebesgue Differentiation Theorem or the Fine Limit Theorem force consistency in this assignment. The zero sets on the boundary are then used in constructing approach neighborhoods which are level sets in the disk. These neighborhoods form the coarsest filters for which a Fatou boundary limit theorem holds and the required zero limits achieved. The construction is new for the unit disk, but it is also valid for very general settings. It shows that any limit theorem for positive harmonic functions can be replaced with one which is at least as good (in terms of the coarseness of the filters) where the approach neighborhoods are generated by level sets of harmonic functions.

Sturdy Harmonic Functions and their Integral Representations

Sturdy harmonic functions constitute all but the least tractable of the positive harmonic functions in potential-theoretic settings. They are the uniform limits on compact sets of positive, bounded harmonic functions and are also produced by a simple integral representation on the boundary of a natural compactification of the space on which they are defined. The boundary of that compactification is metrizable, and more regular for the Dirichlet problem, in general, than is the Martin boundary if that boundary is even defined in the setting.

Approximation by Continuous Potentials

In this note we improve theorems in [1] and [2] dealing with approximation of (super)harmonic functions by continuous potentials. That is, we intend to show that for every finely open set G of a balayage space (X, W) there exists a continuous potential q e P such that

Simplicial cones in potential theory

S(G) = \overline {P + \mathbb{R}q} ,H(G) = \overline {H(q)}

Cones of hyperharmonic functions

.

The optimal differentiation basis and liftings of

The theory of harmonic spaces was mainly established with the aim to generalize and unify results and methods of classical potential theory for application to an extended class of elliptic and parabolic differential equations of second order. Originally, the theory started with a sheaf of vector spaces of real continuous functions on a locally compact space, playing the role of the sheaf of solutions of a partial differential equation. A convergence property, the boundary minimum principle, the local solvability of the Dirichlet problem are supposed to hold. The most important type of a harmonic space is a Bauer space which is introduced in section 1. In our terminology a Bauer space is locally a harmonic space having a base of regular sets. Semi-elliptic differential operators are treated in section 2. In section 3 we present J. M. BONY’s result that a Bauer space whose harmonic functions are smooth is generated by such a differential operator. Section 4 prepares the material (Sobolev spaces, weak solutions, etc.) needed in section 5 to show that elliptic-parabolic differential operators generate Bauer spaces. Besides the deep result of L. HORMANDER on the hypoellipticity of such operators the theory is completely selfcontained. For sake of simplicity we mostly assume that the constant function 1 is harmonic.