Asymptotically holomorphic functions and certain of their applications. (English) Zbl 0744.30038

Volberg, Alexander L.

Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. II, 959-967 (1991).

[For the entire collection see Zbl 0741.00020.]
The classical Phragmén-Lindelöf theorems are of the form: If \(f(z)\in H(D)\) satisfies certain conditions on the boundary \(\partial D\) of \(D\), then either \(f(z)\) is bounded on \(D\) or \(f(z)\) must be rather big somewhere in \(D\). The author’s theory of “asymptotically holomorphic” \((AH)\) functions gives analogues of these theorems for the much wider class of \(AH\)-functions.
An \(AH\)-function \(f(z)\) defined in \(D\) is a \(C^ 1\)-function subject to two types of conditions: 1) growth conditions on \(f\) and 2) a condition enforcing that \(\overline\partial f\) tends to zero at a prescribed (rather fast) rate as \(z\to\partial D\). Three precise theorems are stated for the cases of \(D\)=disk, \(D\)=half-plane, \(D\)=punctured disk.
Three important applications of these theorems are explained by the author. They are 1. Far-reaching generalizations of Beurling’s well-known characterization of the shift-invariant subspaces of the Hardy space \(H^ 2(U)\).
2. The determination of the closure of \(\mathbb{C}[z]\) in the Banach space \(L^ p(\mu)\) \((1\leq p<\infty)\) with the norm \(\| g\|^ p=\int| g|^ pd\mu\), where \(\mu\) is a positive Borel measure with compact support [J. E. Thomson, Ann. Math., II. Ser. 133, 477- 507 (1991; Zbl 0736.41008)]. 3. Proof that the differential equation \(\dot x=\alpha(x,y)\), \(\dot y=\beta(x,y)\) (\(\alpha\), \(\beta\) quasi- analytic functions subject to growth conditions) has only a finite number of limit cycles in any compact region of the plane, provided that all singular points are non-degenerate [A. L. Volberg, Publ. Math. Orsay, Sémin. Anal. Harmonique 1989/90, 152-171 (1990; Zbl 0719.34052)].

Reviewer: W.H.J.Fuchs (Ithaca)

MSC:

30G30

Other generalizations of analytic functions (including abstract-valued functions)

Keywords:

Phragmén-Lindelöf theorems; Hardy space; limit cycles