Characterization of Quasi-units in Terms of Equilibrium Potentials'

In the cone of nonnegative superharmonic functions on a bounded euclidean region Q, quasi-units were introduced as those elements invariant under the infinitesimal generator of the fundamental one-parameter semigroup of operators Sx ( X > 0). All harmonic measures and capacitary potentials are quasi-units, but the latter class has more extensive closure properties. Quasi-units arise naturally under various operations of classical potential theory and have important applications, for example in proving that the convex set of Green's potentials u of positive mass distributions on £2 with u « 1 has as its extreme points precisely the capacitary potentials. Some new properties of quasi-units are developed here. In particular, it is shown that quasi-units can be characterized as limits of increasing sequences of continuous equilibrium potentials for which the equilibrium values tend to 1. 1. Let 0) on %. Here, and throughout the discussion, e will be fixed as any nonzero element of %. In the most classical situations e is taken as the constant function e = 1, and it Received by the editors February 24, 1982 and, in revised form, November 16, 1982. Presented to the Society, April 24, 1981. 1980 Mathematics Subject Classification. Primary 31A15, 31B15, 31D05. 1 Research supported in part by the National Science Foundation. ©1983 American Mathematical Society 0002-9939/83 $1.00 + $.25 per page 267 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 268 MAYNARD ARSOVE AND HEINZ LEUTWILER will be convenient in the general case to regard e as a "unit element". Relative to this element, the operators SXe are defined by setting (1.1) SXeu = R(u Xe) + (uG%), where R denotes the regularized reduced function operator. That is, for 0 any nonnegative function on Q, R }. We now define an e-quasi-unit as any element u of % such that the equality (1.2) SXeu = (\-X)u holds for all X on the interval (0.1). It is easy to see that all e-quasi-units u satisfy u < e and that (1.2) is actually equivalent to the pair of conditions (1.3) u < e and SXeu>(l-X)u. It will be recalled that, for A a compact subset of the region fi and X a positive constant, the equilibrium potential for the set A with equilibrium value X is just XRf. Here we use A*f in the usual way to denote R(x¡.-), where \e 's tne characteristic function of E. It is readily verified that equilibrium potentials, defined in this way, are necessarily Green's potentials over Q. In what follows, we shall have use for a natural extension of the concept of an equilibrium potential, in which the constant function 1 is replaced by our generalized unit element e, and the compact set A is replaced by an arbitrary subset E of £2. We shall refer to XRb; as a generalized equilibrium function for the set E, having equilibrium value X (relative to e). Our main result can now be stated, in the classical case of e = 1, as follows: a function u on S2 is a 1-quasi-unit if and only if it is the limit of an increasing sequence of continuous equilibrium potentials with equilibrium values tending to 1. More generally, a corresponding characterization holds for e-quasi-units, namely Theorem 1.1. Let e be any continuous nonzero function in %. Then a function u o« ñ is an e-quasi-unit if and only if it is the limit of an increasing sequence {XnR^") of continuous generalized equilibrium potentials with all Kn compact and Xn -* 1. Before we turn to the proof of Theorem 1.1 and related results, it may be of interest to have some background information on the development of the quasi-unit concept and the scope of the resulting theory. The notion of a quasi-unit was introduced in [2] in the abstract setting of a strongly superharmonic cone. There it was shown that the operators SXe form a one-parameter semigroup, and quasi-units were defined as those elements invariant under the infinitesimal generator of the semigroup. Our defining property (1.2) appears in [2] as one of the characterizations of quasi-units, and it is also equivalent to the defining property adopted in the algebraic setting of [3]. As noted in [2] (see also [5]), property (1.2) will hold for all À on (0,1) if it holds for one X on that interval. A large number of closure properties of quasi-units are established in [2]-[5], along with various characterizations and other properties. Quasi-units are applied in [2] to arrive at an extension of the classical Freudenthal spectral theorem, and further applications can be found in [5], where quasi-units are used to obtain characterizations of the extreme points of certain License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use QUASI-UNITS IN TERMS OF EQUILIBRIUM POTENTIALS 269 convex sets in %. In particular, it is shown in [5] that the convex set of Green's potentials in % which are bounded above by 1 has as its extreme points precisely the capacitary potentials in %. We note also that [5] treats the theory of quasi-units in the classical and axiomatic settings independently of the algebraic theory in [2] and [3] and makes use of the theory of finitely harmonic functions (as developed by B. Fuglede [7]) to obtain new results in those settings. It should be remarked that a discussion of the quasi-units of K. Yosida can be found in his book [8]. 2. Some properties and characterizations of quasi-units. We proceed to derive some new properties and characterizations of quasi-units that hold in the present setting. Throughout the present section % can, in fact, be taken as the set of all nonnegative superharmonic functions on an arbitrary harmonic space, and Theorem 2.3 even remains valid in the abstract setting of [2]. For u an e-quasi-unit and 0 < X < 1, it is easy to see that (2.1) ■ Xu ^ XR[^Xe] < u and, similarly, (2.2) Xu^XR[u>Xe]*Zu. For example, to prove (2.1), one has only to observe that (u-Xe)+ ^(\ -X)eXlu9Xe]