Multiplier theorems for star-invariant subspaces
Anton Baranov (Joint work with A. Borichev and V. Havin) ABSTRACT
Let Θ be an inner function in the upper half-plane and letKΘ=H2⊖ΘH2 be the associated star-invariant subspace of the Hardy classH2. A nonnegative functionwon the real line is said to be an admissible majorant forKΘif there is a non-zero functionf ∈ KΘ such that |f| ≤w a.e. on R. In other words, there exists a ”multiplier”f ∈KΘ such thatf w−1 ∈L∞(R). In an important particular case Θ(z) = exp(iaz), a >0, the subspaceKΘ essentially coincides with the Paley-Wiener spaceP Wa and admissible majorants are described by the famous Beurling-Malliavin Multiplier Theorem.
We consider the case when Θ is a meromorphic Blascke product with the zerosznand, thus,KΘis the space of square summable meromorphic functions with the poles at the pointszn. Making use of a recent approach of V.Havin and J. Mashreghi, we study the relations between the distribution of the zeros of a Blaschke product Θ and the class of admissible majorants for the space KΘ. Also we consider the problem of two-sided estimates (that is, estimates of the formC1w≤ |f| ≤C2w) for the classical Paley-Wiener spaces.
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