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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Weighted Bernstein-type inequalities, and embedding theorems for the model subspaces


Author: A. D. Baranov
Translated by: the author
Original publication: Algebra i Analiz, tom 15 (2003), nomer 5.
Journal: St. Petersburg Math. J. 15 (2004), 733-752
MSC (2000): Primary 47B32, 30B50
Published electronically: July 29, 2004
MathSciNet review: 2068792
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Abstract: Weighted estimates are obtained for the derivatives in the model (shift-coinvariant) subspaces $K^p_{\Theta}$, generated by meromorphic inner functions $\Theta$ of the Hardy class $H^p(\mathbb{C} ^+)$. It is shown that the differentiation operator acts from $K^p_{\Theta}$ to a space $L^p(w)$, where the weight $w$ depends on the function $\vert\Theta'\vert$, the rate of growth of the argument of $\Theta$ along the real line.

As an application of the weighted Bernstein-type inequalities, new Carleson-type theorems on embeddings of the subspaces $K^p_{\Theta}$ in $L^p(\mu)$ are proved. Also, results on the compactness of such embeddings are obtained, and properties of measures $\mu$ for which the norms $\Vert\cdot\Vert _{L^p(\mu)}$ and $\Vert\cdot\Vert _p$ are equivalent on a given model subspace $K^p_{\Theta}$, are established.


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Additional Information

A. D. Baranov
Affiliation: St. Petersburg State University, Universitetskii Prospekt 28, Petrodvorets, St. Petersburg, 198504, Russia
Email: d.baranov@pop.ioffe.rssi.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-04-00829-5
PII: S 1061-0022(04)00829-5
Keywords: Hardy class, inner function, shift-coinvariant subspace, Bernstein-type inequality
Received by editor(s): March 6, 2003
Published electronically: July 29, 2004
Additional Notes: The work was partially supported by RFBR grant no. 03-01-00377 and by the grant for Leading Scientific Schools no. NSH-2266.2003.1.
Article copyright: © Copyright 2004 American Mathematical Society