Uniqueness theorems for almost analytic functions.

*(English. Russian original)*Zbl 0725.30038
Leningr. Math. J. 1, No. 1, 157-191 (1990); translation from Algebra Anal. 1, No. 1, 146-177 (1989).

Recently Vol’berg proved a uniqueness theorem for “almost analytic” functions that generalizes in some sense the well-known theorems of Jensen, Levinson and Cartwright, and Beurling [see A. L. Vol’berg, Dokl. Akad. Nauk SSSR 265, No.6, 1297-1302 (1982; Zbl 0518.30034), A. L. Vol’berg and the reviewer, Math. USSR, Sb. 58, No.2, 337-349 (1987; Zbl 0644.30020)]. Let \(f\in L^ 1({\mathbb{T}})\) (\({\mathbb{T}}\) denotes the unit circle) satisfy \(| \hat f(n)| <e^{-M(| n|)}\) for \(n<0\) with
\[
\int^{\infty}_{1}(M(x)/x^ 2)dx=\infty
\]
and M satisfying some regularity conditions. Then \(\int_{{\mathbb{T}}}\log | f| dm>-\infty\) unless \(f\equiv 0\). The theorems of Levinson and Cartwright [resp. of Beurling] state only that f cannot vanish on a nontrivial arc [resp. on a set of positive measure] unless it vanishes identically. The regularity conditions in Vol’berg’s theorem are more restrictive than those in the Levinson-Cartwright and Beurling theorems. By a construction of E. M. Dyn’kin [Mat. Sb., Nov. Ser. 89(131), 182-190 (1972; Zbl 0251.30033)] one can replace “almost analytic” functions on \({\mathbb{T}}\) by bounded functions \(F\in C^ 1({\mathbb{D}})\) (\({\mathbb{D}}\) denotes the unit disc) with \(| {\bar \partial}F(z)| \leq \omega (1-| z|)\), where
\[
(*)\quad \int_{0}\log \log (1/\omega (t))dt=+\infty
\]
and \(\omega\) satisfies some regularity conditions. The boundary functions of such \(| F|\) have integrable logarithm unless they are identically zero.

In the present paper the authors consider, instead of bounded functions F, functions with restricted growth: \[ (**)\quad | F(z)| \leq c_ 1\{\omega (c(1-| z|))\}^{-1}, \] for some \(c_ 1\), \(c>0\). If, also, for all c there exists \(c_ 1\) such that \(| {\bar \partial}F(z)| \leq c_ 1\omega (c(1-| z|))\) then various uniqueness theorems hold (\(\omega\) satisfies (*) and regularity conditions). For example, if such a function F has vanishing nontangential boundary values on a set of positive measure then F is rapidly decreasing: for all c there exists \(c_ 1\) such that \(| F(z)| \leq c_ 1\omega (c(1-| z|)).\)

Also, Vol’berg’s theorem is generalized for functions \(F\in C^ 1({\mathbb{D}})\) which satisfy certain growth conditions (stronger than (**)) instead of being bounded. In this case boundary values may not exist on \({\mathbb{T}}\). The corresponding integrability condition for the logarithm takes the form \[ \liminf_{j\to \infty}\int_{{\mathbb{T}}}\log | F(r_ je^{i\theta})| d\theta >-\infty \] for some sequence \(r_ j\uparrow 1.\)

The paper contains many other interesting results. At the end of the paper the growth and regularity conditions are discussed. As a rule the growth conditions cannot be considerably weakened unless the results become wrong. Also the regularity condition in Vol’berg’s theorem on the summability of the logarithm cannot be much weakened unless the result becomes wrong (this concerns the regularity condition proposed by J. E. Brennan [in Complex analysis I, Proc. Spec. Year, College Park/Md. 1985-86, Lect. Notes Math. 1275, 31-43 (1987; Zbl 0646.30034)] which differs slightly from that in the work of Vol’berg and the reviewer [op. cit.] but can be shown to be sufficient, essentially using the methods of the latter work).

In the present paper the authors consider, instead of bounded functions F, functions with restricted growth: \[ (**)\quad | F(z)| \leq c_ 1\{\omega (c(1-| z|))\}^{-1}, \] for some \(c_ 1\), \(c>0\). If, also, for all c there exists \(c_ 1\) such that \(| {\bar \partial}F(z)| \leq c_ 1\omega (c(1-| z|))\) then various uniqueness theorems hold (\(\omega\) satisfies (*) and regularity conditions). For example, if such a function F has vanishing nontangential boundary values on a set of positive measure then F is rapidly decreasing: for all c there exists \(c_ 1\) such that \(| F(z)| \leq c_ 1\omega (c(1-| z|)).\)

Also, Vol’berg’s theorem is generalized for functions \(F\in C^ 1({\mathbb{D}})\) which satisfy certain growth conditions (stronger than (**)) instead of being bounded. In this case boundary values may not exist on \({\mathbb{T}}\). The corresponding integrability condition for the logarithm takes the form \[ \liminf_{j\to \infty}\int_{{\mathbb{T}}}\log | F(r_ je^{i\theta})| d\theta >-\infty \] for some sequence \(r_ j\uparrow 1.\)

The paper contains many other interesting results. At the end of the paper the growth and regularity conditions are discussed. As a rule the growth conditions cannot be considerably weakened unless the results become wrong. Also the regularity condition in Vol’berg’s theorem on the summability of the logarithm cannot be much weakened unless the result becomes wrong (this concerns the regularity condition proposed by J. E. Brennan [in Complex analysis I, Proc. Spec. Year, College Park/Md. 1985-86, Lect. Notes Math. 1275, 31-43 (1987; Zbl 0646.30034)] which differs slightly from that in the work of Vol’berg and the reviewer [op. cit.] but can be shown to be sufficient, essentially using the methods of the latter work).

##### MSC:

30G30 | Other generalizations of analytic functions (including abstract-valued functions) |