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Interpolating sequences on analytic Besov type spaces. (English) Zbl 1213.30069

Let \(p\) and \(s\) be real numbers such that \(p>1\) and \(1/s>1\). The space \(B_p(s)\) is formed by those functions \(f\) holomorphic on the open unit disk \(\mathbb{D}\) such that \[ \|f\|^p=|f(0)|^p+\int_\mathbb{D}|f'(z)|^p(1-|z|^2)^{s+p-2}\mathrm{d}A(z) \] is finite (\(A\) denotes the normalized area measure). This identity defines a Banach norm \(\| \cdot\|\) on this space. These are the Besov type spaces referred to in the title of the present paper.
A sequence \((z_n)\) in \(\mathbb{D}\) is an interpolating sequence for (the multipliers of) \(B_p(s)\) if every bounded complex sequence can be interpolated along \((z_n)\) by a multiplier of \(B_p(s)\). The authors prove that \((z_n)\) is interpolating if and only if it satisfies the following condition, abbreviated by (CS): the atomic measure \(\mu=\sum_n(1-|z_n|^2)\delta_{z_n}\) is such that \(B_p(s)\) is boundedly contained in \(L^p(\mu)\) (such a measure \(\mu\) is called a Carleson measure for \(B_p(s)\)).
As a result, the authors obtain three different conditions characterizing the interpolating sequences for these spaces, which were previously known to be equivalent (W. Cohn [Complex Variables, Theory Appl. 22, No.1–2, 35–45 (1993; Zbl 0793.30043)], N. Arcozzi, R. Rochberg and E. Sawyer [Mem. Am. Math. Soc. 859, (2006; Zbl 1112.46027)]).
If (CS) is satisfied, the authors describe two different ways to construct interpolating multipliers. The first uses a solution of the \(\bar\partial\)-problem, and the second follows the technique of J. Earl [J. Lond. Math. Soc., II. Ser. 2, 544–548 (1970; Zbl 0195.35702)] and the fact proved by I. Verbitsky [Sib. Math. J. 26, 198–216 (1985; Zbl 0591.46025)] that \((z_n)\) satisfies the condition (CS) if and only if the Blaschke product associated to it is a multiplier of \(B_p(s)\) (the authors provide an elementary proof of this fact as well).
As a by-product of their solution for the \(\bar\partial\)-problem, related to the techniques introduced in a famous paper by P. Jones [Acta Math. 150, 137–152 (1983; Zbl 0516.35060)], the authors obtain a new proof of the Corona Theorem for the multipliers of \(B_p(s)\) (V. Tolokonnikov [Transl., Ser. 2, Am. Math. Soc. 149, 61–95 (1991; Zbl 0765.46041)]).

MSC:

30E05 Moment problems and interpolation problems in the complex plane
30H25 Besov spaces and \(Q_p\)-spaces
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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