Peak Points for Pseudoconvex Domains: A Survey

Abstract

Let D be a domain in C n with smooth (that is, C ∞ ) boundary. If 0 ≤ α ≤ ∞ we denote by A α (D) the space of functions holomorphic on D and of class C α on D. We write A(D) for A 0 (D) and A ω (D) for the space of functions holomorphic on a neighborhood of D. We say that a point p ∈ ∂D is a peak point relative to D for A α (D) if there exists a function f ∈ A α (D) so that f (p) = 1 and |f | < 1 on D \ {p}. We call f a peak function. This condition is clearly equivalent to the existence of a strong support function, that is, a function g ∈ A α (D) so that g(p) = 0 and Re g > 0 on D \ {p}. We say that p ∈ ∂D is a local peak point forWe want to determine whether boundary points are peak points. Finding a peak function can be thought of as giving a quantitative converse to the maximum modulus principle. Now observe that if f is a peak function at p relative to D then 1/(1 − f ) is a holomorphic function on D with no holomorphic extension past p. Thus if every boundary point of D is a peak point then D is a domain of holomorphy. In light of this fact, if we want to determine whether boundary points are peak points it makes sense to restrict our attention to domains of holomorphy. By the solution of the Levi problem these are the (Levi) pseudoconvex domains. Here is another observation: If there is a complex disc in the boundary of D then (by the maximum modulus principle) no point in the relative interior of that disc can be a peak point for A(D). A natural condition in this context is that D be of finite type at the point p in question, in the sense of D'Angelo: There is a finite upper bound on the order of contact of nontrivial complex analytic varieties with the boundary at p. The infimum of all such upper bounds is called the type at p. We formulate the major open question in terms of this notion.

Plurisubharmonic polynomials and bumping

“…Result 2.1 (Lemma 4.2, [11]). Let P be a homogeneous, plurisubharmonic, nonpluriharmonic polynomial in C n , n ≥ 2.…”

“…, z n−1 ), centered at ζ, such that Ω V ζ = (w, z) ∈ V ζ : Re(w) + P 2k (z) + O(|z| 2k+1 , |wz|, |w| 2 ) < 0 , where P 2k is a plurisubharmonic polynomial in C n−1 that is homogeneous of degree 2k. The first result in C n , n ≥ 3, to address (*) is due to Noell [11]. He showed that if P 2k is plurisubharmonic and is not harmonic along any complex line through 0, then Ω can be bumped homogeneously to order 2k around 0 ∈ C n .…”

Peak sets and boundary interpolation sets for the unit disc: a survey

“…For results about peak sets for Lipschitz classes in , see Ababou–Boumaaz [1], Saerens [53], Cascante [14], and Noell [42].…”

q‐pseudoconvex and q‐holomorphically convex domains

“…The following theorem by Rossi [, Theorem 4.4], also mentioned in Noell's survey [, Remark 2.4 (2)], gives a sufficient condition for the existence of peak functions. Theorem If D is an open subset of a Stein manifold and has a basis of strictly pseudoconvex domains, then every local peak point for is a global peak point for .…”