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Traces of bounded analytic functions on finite unions of Carleson sets. (Russian. English summary) Zbl 0533.30033

Let \(H^{\infty}\) denote the space of all bounded analytic functions on the open unit disk \(D=\{z: | z|<1\}\). For a subset \(\Lambda\) of D, \(\ell^{\infty}(\Lambda)\) denotes the set of all complex-valued bounded functions on \(\Lambda\). Then the well-known result of Carleson states that \(H^{\infty}| \Lambda =\ell^{\infty}(\Lambda)\) if and only if \(\Lambda\) satisfies the Carleson condition: there exists a constant \(\delta>0\) such that \(\prod_{i\neq n}(| \lambda_ i- \lambda_ n| /| 1-{\bar \lambda}_ n\lambda_ i|)\geq \delta\) for all n, where \(\Lambda =\{\lambda_ n: n=1,2,...\}\). In the paper under review the author characterizes such a \(\Lambda\), which can be expressed as the union of finitely many sequences satisfying the Carleson condition. Given any subset \(\Lambda\) of D and any function w on \(\Lambda\), we define functions \(\Delta^ kw\), \(k=0,1,...\), as follows: \((1)\quad \Delta^ 0w(\lambda)=w(\lambda)\) for any \(\lambda\in \Lambda\); (2) when \(\Delta^{k-1}w\) is defined as a function on the product set \(\Lambda^{(k)}=\Lambda \times...\times \Lambda\) (k-tuple), \(\Delta^ kw(\lambda ',\lambda_ k,\lambda_{k+1}) (\lambda '\in \Lambda^{(k- 1)}\), \(\lambda_ k,\lambda_{k+1}\in \Lambda)\) is defined to be the quotient \((\Delta^{k-1}w(\lambda ',\lambda_{k+1})-\Delta^{k- 1}w(\lambda ',\lambda_ k))/((\lambda_ k-\lambda_{k+1})/(1-{\bar \lambda}_ k\lambda_{k+1})).\) Then \(X_ n(\Lambda)\) is defined as the set of all functions \(w(\lambda)\) on \(\Lambda\) such that \(\Delta^ nw\) is bounded on \(\Lambda^{(n+1)}\). Theorem: If \(\Lambda\subset D\) is the union of n sequences satisfying the Carleson condition, then \(H^{\infty}| \Lambda =X_ n(\Lambda).\) The case \(n=1\) is exactly the theorem of Carleson. The case \(n=2\) was discussed by A. Dufresnoy [J. Funct. Anal. 21, 245-285 (1976; Zbl 0318.46062)].
Reviewer: M.Hasumi

MSC:

30D55 \(H^p\)-classes (MSC2000)
30E05 Moment problems and interpolation problems in the complex plane
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces

Citations:

Zbl 0318.46062
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