Construction of P.S.H. functions on weakly pseudoconvex domains

Abstract

Let Q c C be a smoothly bounded pseudoconvex domain, and let A(Q) denote the functions holomorphic on Q and continuous on Sl. A point p e aQ is a peak point if there is a function f e A(Q) such that f(p) = 1 and If(z)I 2) were obtained by Range [19], [20] and Hakim and Sibony [12]. These results do not extend automatically to arbitrary weakly pseudoconvex boundary points of Q, even if the boundary is real analytic. An example of Kohn and Nirenberg [17] shows that a p e aQ does not always have a separating function which is real analytic at p. Similarly, this example does not have a peak function which is e' at p (see Fornaess [6]). Here we show how peak functions and separating functions may be constructed for a large class of weakly pseudoconvex points on domains in C*. The results are most complete for domains in C2, so we state them here for that case.