Minimal positive harmonic functions

Introduction. One may ask how great generality in a domain is to be permitted if we are to have for this domain a formula possessing the more significant features of the Poisson-Stieltjes integral formula for the circle or the sphere('). Even if there is agreement as to what the more important consequences of the formula are, there are two approaches, differing not so much in content as in emphasis, along which partial answers to the question lie. The first consists in determining hypotheses, as weak as possible, upon a domain under which all, or substantially all, of the important features of the formula admit of extension. The second consists in attempting to determine for each of the important consequences of the formula the class of domains for which it holds. While this sounds very much like the distinction between obtaining sufficient and obtaining necessary conditions for an extension of the formula together with all of its important features, actually it goes a little deeper, since the second viewpoint involves implicitly the notion that what a significant extension of the formula is may depend upon what it is going to be used for. It is a particular consideration from the viewpoint of the second approach which leads to the concept of a minimal positive harmonic function with which we are concerned in the present article. A function positive and harmonic in a given domain we shall call minimal(2)-for this domain-if it dominates there no positive harmonic function except for its own constant submultiples. An important instance of this kind of function occurs in connection with the principle of Picard(3), whose relation