Polynomial convexity

Let denote the set of holomorphic polynomials on (cf. also Analytic function). Let be a compact set in and let be the sup-norm of on . The set

is called the polynomially convex hull of . If one says that is polynomially convex.

An up-to-date (as of 1998) text dealing with polynomial convexity is [a3], while [a13] and [a27] contain some sections on polynomial convexity, background and older results. The paper [a24] is an early study on polynomial convexity.

Polynomial convexity arises naturally in the context of function algebras (cf. also Algebra of functions): Let denote the uniform algebra generated by the holomorphic polynomials on with the sup-norm. The maximal ideal space of is the set of homomorphisms mapping onto , endowed with the topology inherited from the dual space . It can be identified with via

Moreover, if is any finitely generated function algebra on a compact Hausdorff space, then is isomorphic to , where for one can take the joint spectrum of the generators of (cf. also Spectrum of an operator).

By the Riesz representation theorem (cf. Riesz theorem) there exists for every at least one representing measure , that is, a probability measure on such that

One calls a Jensen measure if it has the stronger property

It can be shown that for each there exists a Jensen measure . See e.g. [a27].

For compact sets in one obtains by "filling in the holes" of , that is, , where is the unbounded component of . In , , there is no such a simple topological description.

Early results on polynomial convexity, cf. [a13], are

Oka's theorem: If is a polynomially convex set in and is holomorphic on a neighbourhood of , then can be written on as a uniform limit of polynomials. Cf. also Oka theorems.

Browder's theorem: If is polynomially convex in , then for .

Here, is the th Čech cohomology group. More recently (1994), the following topological result was obtained, cf. [a9], [a3]:

Forstnerič' theorem: Let be a polynomially convex set in , . Then

and

Here, denotes the th homology group of with coefficients in an Abelian group and is the th homotopy group of .

One method to find is by means of analytic discs. Let be the unit disc in and let be its boundary. An analytic disc is (the image of) a holomorphic mapping such that is continuous up to . Similarly one defines an -disc as a bounded holomorphic mapping . Its components are elements of the usual Hardy space (cf. Hardy spaces).

Now, let be compact in and suppose that for some analytic disc . Then by the maximum principle applied to for polynomials . The same goes for -discs whose boundary values are almost everywhere in . One says that the disc is glued to . Next, one says that has analytic structure at if there exists a non-constant analytic disc such that and the image of is contained in .

It was a major question whether always has analytic structure. Moreover, when is obtained by glueing discs to ? One positive result in this direction is due to H. Alexander [a1]; a corollary of his work is as follows: If is a rectifiable curve in , then either and , or is a pure -dimensional analytic subset of (cf. also Analytic set). If is a rectifiable arc, is polynomially convex and .

See [a1] for the complete formulation. Alexander's result is an extension of pioneering work of J. Wermer, cf. [a30], E. Bishop and, later, G. Stolzenberg [a26], who dealt with real-analytic, respectively , curves. Wermer [a29] gave the first example of an arc in that is not polynomially convex, cf. [a3]. However, Gel'fand's problem (i.e., let be an arc in such that ; is it true that ?) is still open (2000). Under the additional assumption that its projections into the complex coordinate planes have -dimensional Hausdorff measure , the answer is positive, see [a3].

F.R. Harvey and H.B. Lawson gave a generalization to higher-dimensional , cf. [a12], which includes the following.

Let . If is a -dimensional submanifold of and at each point of the tangent space to contains a -dimensional complex subspace, then is the boundary of an analytic variety (in the sense of Stokes' theorem).

Another positive result is contained in the work of E. Bedford and W. Klingenberg, cf. [a4]: Suppose is the graph of a -function over the boundary of a strictly convex domain . Then is the graph of a Lipschitz-continuous extension of on . Moreover, is foliated with analytic discs (cf. also Foliation).

The work of Bedford and Klingenberg has been generalized in various directions in [a16], [a21] and [a7]. One ingredient of this theorem is work of Bishop [a5], which gives conditions that guarantee locally the existence of analytic discs with boundary in real submanifolds of sufficiently high dimension. See [a11], [a32] and [a15] for results along this line.

A third situation that is fairly well understood is when is a compact set fibred over , that is, is of the form , where is a compact set in depending on .

In this case the following is true: Let be a compact fibration over the circle and suppose that for each the fibre is connected and simply connected. Then is the union of graphs , where and the boundary values are in for almost all .

Of course, it is possible that is empty. The present theorem is due to Z. Slodkowski, [a22], earlier results are in [a2] and [a10]. Slodkowski proved a similar theorem in under the assumption that the fibres are convex, see [a23].

Despite these positive results, in general need not have analytic structure. This has become clear from examples by Stolzenberg [a25] and Wermer [a31]. Presently (2000) it is not known whether has analytic structure everywhere if is a (real) submanifold of , nor is it known under what conditions is obtained by glueing discs to .

However, it has been shown that in a weaker sense there is always a kind of analytic structure in polynomial hulls. Let denote Lebesgue measure on the circle and let denote the push-forward of under a continuous mapping . Let also be a compact set in . The following are equivalent:

1) and is a Jensen measure for supported on ;

2) There exists a sequence of analytic discs such that and in the weak- sense (cf. also Weak topology).

This was proved in [a6]; [a8] and [a20] contain more information about additional nice properties that can be required from the sequence of analytic discs. Under suitable regularity conditions on , it is shown in [a19] that consists of analytic discs such that has Lebesgue measure arbitrary close to .

Another problem is to describe assuming that and given reasonable additional conditions on . In particular, when can one conclude that ? Recall that a real submanifold of is totally real at if the tangent space in does not contain a complex line (cf. also CR-submanifold). The Hörmander–Wermer theorem is as follows, cf. [a14]: Let be a sufficiently smooth real submanifold of and let be the subset of consisting of points that are not totally real. If is a compact polynomially convex set that contains an -neighbourhood of , then contains all continuous functions on that are on the uniform limit of functions holomorphic in a neighbourhood of .

See [a17] for a variation on this theme. One can deal with some situations where the manifold is replaced by a union of manifolds; e.g., B.M. Weinstock [a28] gives necessary and sufficient conditions for any compact subset of the union of two totally real -dimensional subspaces of to be polynomially convex; then also . See also [a18].

References

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