Positivity 7: 355–387, 2003.
© 2003 Kluwer Academic Publishers. Printed in the Netherlands.
Sturdy Harmonic Functions and their Integral
and PETER A. LOEB
Fachbereich Mathematik, Universität Frankfurt, D-60054 Frankfurt am Main, Germany.
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Received 19 March 2001; accepted 1 July 2002
Abstract. 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 bound-
ary 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.
2000 Mathemtics Subject Classiﬁcation: Primary 31A10, 31A20, 31C35, 31D05, 46A55; Second-
Key words: harmonic functions, integral representations, Dirichlet problem, Martin boundary, Cho-
In classical potential theory, R. S. Martin  constructed what is now called
the Martin compactiﬁcation of a Euclidean domain X. His aim was to obtain an
integral representation of all positive harmonic functions. This representation can
now be considered as an example of Choquet’s integral representation of compact
convex sets in a locally compact Hausdorff space. Unfortunately, as shown in
, the boundary of Martin’s compactiﬁcation, i.e., the Martin boundary, may
have irregularities with respect to the Dirichlet problem that form a set of positive
harmonic measure. In  and , the second author of this paper constructed
a new compactiﬁcation
X of X that in general is better than the Martin boundary
for solving the Dirichlet problem. When using
X, some special positive harmonic
functions may not have an integral representation. On the other hand, the boundary
X enjoys nice regularity properties with respect to the Dirichlet problem. In
addition, the construction of
X works in the framework of harmonic spaces; it can
therefore be applied to a wide class of linear, elliptic and parabolic second order