Psh Functions Research Articles

In the past few years, several capacitary set functions have been introduced in connection with the study of analytic and plurisubharmonic functions of several variables (11-5, 9-13]). There are many different such functions and the relationship between them is not at all clear. We shall consider here the relative capacity of Bedford and Taylor, and the capacity defined in terms of certain Tchebycheff constants as studied by Zaharjata [13] and Alexander [1]. (See Sect. 2 for the definitions.) We show that these capacities are essentially the same. Our main result, Theorem 2.1, gives the quantitative relationship between them. It happens that the relationship between the two capacities is closely connected with a result of Josefson [8] on the equivalence of locally and globally pluripolar sets. The quantitative estimates of Theorem 2.1 allow us to give in Sect. 4 a new proof of Josefson's lemma about normalized polynomials which are very small on the sets where a given plurisubharmonic function is nearly o e . Josefson's own proof is a direct construction; our original motivation was to give a proof based on capacity notions. It turns out that the Tchebycheff polynomials themselves already do the job. E1 Mir [7] has obtained an extension of Josefson's theorem. Given a function v psh on an open set he obtains a global psh function u of small growth at infinity such that u is dominated by a function h(v) of v. His choice of h is essentially h(x)= l o g [xl. Thus he gets u < l o g Iv[ and so the value of the global function u is o e whenever v = oe. Our methods give a short proof of this; in fact, they apply to functions h satisfying