Bliedtner-Hansen lemma

Let $A$ be a subset of a Euclidean space ${\bf R} ^ { n }$, $n \geq 3$, and let $s$ be a positive superharmonic (lower semi-continuous) function on ${\bf R} ^ { n }$ (cf. also Harmonic function; Semi-continuous function). The balayage $\hat { R } _ { S } ^ { A }$ of $s$ on $A$ is defined as the greatest lower semi-continuous minorant of the function

\begin{equation*} R _ { s } ^ { A } : = \operatorname { inf } \left\{ t : \begin{array} { l } \ {t \ \text{superharmonic on}\ \mathbf{R}^{n} , } \\ { t \geq s \ \text{on} \ A } \end{array} \right\}. \end{equation*}

Balayaged functions were introduced in classical potential theory by M. Brelot and play an important role. Given any set $A \subset \mathbf{R} ^ { n }$ and any $x \in \mathbf{R} ^ { n }$, there is a unique Radon measure $\varepsilon _ { X } ^ { A }$ on ${\bf R} ^ { n }$ such that $\varepsilon _ { x } ^ { A } ( s ) = \widehat { R } _ { s } ^ { A } ( x )$ for any superharmonic function $s$. It can proved that

\begin{equation} \tag{a1} \hat { R } _ { \hat{R} _ { S } ^ { A } } ^ { A } = \hat { R } _ { S } ^ { A } \text { on } {\bf R} ^ { n } \end{equation}

If, now, $U$ is a bounded open subset of ${\bf R} ^ { n }$, $\mathcal{C} U : = \mathbf{R} ^ { n } \backslash U$, and $f$ is a Borel function on the boundary $\partial U$ of $U$, then the function $H _ { f } ^ { U }$: $x \mapsto \varepsilon _ { x } ^ { \mathcal{C}U } ( f )$, which is harmonic on $U$, represents a generalized solution of the classical Dirichlet problem. According to (a1), "the solution of a solution is again a solution" . More precisely, given a (bounded) Borel function $f$ on $\partial U$, then the function $g$: $z \mapsto \varepsilon _ { z } ^ { {\cal C} U } ( f )$, $z \in \partial U$, is a Borel function and $\varepsilon _ { X } ^ { \mathcal{C} U } ( g ) = \varepsilon _ { X } ^ { \mathcal{C} U } ( f )$ for any $x \in U$.

The last assertion, as well (a1), fails to be valid in more general situations such as, for example, the case of solutions of the heat equation. Replace now ${\bf R} ^ { n }$ by an abstract $\mathcal{P}$-harmonic space (in the axiomatic sense of C. Constantinescu and A. Cornea; see [a4]). By this one understands a locally compact space $X$ having a countable base equipped with a sheaf $\mathcal{H}$ of "harmonic" functions. Main model examples of abstract harmonic spaces are described by solutions of the Laplace equation in ${\bf R} ^ { n }$ and of the heat equation in $\mathbf R ^ { n + 1 }$. Notice also that (a1) expresses nothing else than the axiom of polarity, and this axiom is fulfilled for the first model but fails for the second one.

One may introduced on $X$ a class of superharmonic functions in a natural way and, as above, one may define balayages of superharmonic functions. As superharmonic functions are lower semi-continuous only, one defines the fine topology on $X$ as the weak topology on $X$ generated by the family of all superharmonic functions. This new topology was introduced in classical potential theory by Brelot and H. Cartan around 1940 and later on intensively studied, even in the axiomatic setting of harmonic spaces.

As mentioned above, (a1) is no more valid in abstract harmonic spaces without the presence of the axiom of polarity. Nevertheless, the fundamental Bliedtner–Hansen lemma, which is closely related to (a1), allows one to derive strong results in abstract harmonic spaces. It says that $\varepsilon _ { x } ^ { X \backslash V } ( R _ { s } ^ { X \backslash U } ) = R _ { s } ^ { X \backslash U } ( x )$, provided $U$ and $V$ are open in the fine topology, $U$ is a Borel set in the original topology of $X$ and $x \in V \subset U \subset X$.

The powerful result of J. Bliedtner and W. Hansen can be stated in various degrees of generality. It appeared originally in [a1] and can be proved also in the framework of balayage spaces [a2] or even of standard $H$-cones [a3]. Detailed proofs of the Bliedtner–Hansen lemma can be found in [a5].

References

[a1]	J. Bliedtner, W. Hansen, "Cones of hyperharmonic functions" Math. Z. , 151 (1976) pp. 71–87
[a2]	J. Bliedtner, W. Hansen, "Balayage spaces: An analytic and probabilistic approach to balayage" , Universitext , Springer (1986)
[a3]	N. Boboc, Gh. Bucur, A. Cornea, "Order and convexity in potential theory: H-cones" , Lecture Notes Math. , 853 , Springer (1981)
[a4]	C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972)
[a5]	J. Lukeš, J. Malý, L. Zajíček, "Fine topology methods in real analysis and potential theory" , Lecture Notes Math. , 1189 , Springer (1986)