More about Capacity and Excessive Measures

Abstract

In [14] we introduced the set function

$$\Gamma \left( B \right)={{\Gamma }_{m,u}}\left( B \right)=L\left( m,{{P}_{B}}u \right)=L\left( {{R}_{B}}m,u \right)$$

(1.1)

as a “good” capacity. In this paper we develop some additional properties of Γ(B). We are especially interested in obtaining expressions for Γ(B) as a supremum that are analogous to the classical result of de la Vallée-Poussin which states that the (Newtonian) capacity of a Borel set B is the supremum of µ(1) over all measures µ with compact support in B and whose potential is bounded by 1. Results of this character are contained in sections 3 and 4. In section 5 we obtain some expressions for Γ as an infimum. These results are reminiscent of the work of Fuglede [19] and are in some sense dual to the classical result. Section 6 contains additional formulas for Γ^q(B)-the q-capacity of B; that is, the capacity relative to the q-subprocess. Of special interest is (6.7) which shows that q → Γ^q(B) is a subordinator exponent (i.e., has a completely monotone derivative) provided it is finite for one value of q 0 0. Section 2 contains two results in the potential theory of excessive measures that are needed in section 3. But these are of considerable interest in their own right. Theorem 2.18 sheds light on an old problem going back to [2]. See also [17] and [15].

Research supported by NSF Grant DMS-8419377.