INTEGRAL REPRESENTATION OF CONTINUOUS FUNCTIONS(x)

It was shown by F. Riesz [5; 350](2) that every subharmonic function u can be represented as the sum of the potential of its mass distribution plus a harmonic function; the potential appears in the form of a Stieltjes integral (Riesz's theorem is stated in (2.2.1)). We prove that the Stieltjes integral may be replaced by a Lebesgue integral if u is continuous, and if the lower generalized Laplacian of u is less than + <», except possibly on a set of capacity zero (Theorem II). In other words, the above assumptions imply the absolute continuity of the mass distribution associated with u. We obtain this result as a consequence of Theorem I, which deals with the representation of continuous functions in integral form. In another paper, Theorem I will be used in an investigation of the uniqueness theory for Laplace series. The theorem was actually suggested by this application, and is of a type similar to a theorem of Zygmund [9; 276] on the representation of continuous functions of one variable. Our results are stated for the plane, but analogous theorems evidently hold for continuous functions of three or more variables, if the generalized Laplacians are suitably defined (see, for instance, [7]). 1.1. Notation. Let D be a finite plane domain (that is, a connected open set not containing the point at infinity). Let Z be a closed and bounded plane set of capacity zero (see 2.3). Let J(P, r) denote the closed circular disc bounded by the circle C(P, r) with center at P, and radius r. If the function F EL on C(P, r), we put