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

 

The sharp constant in the reverse Hölder inequality for Muckenhoupt weights


Author: V. Vasyunin
Translated by: the author
Original publication: Algebra i Analiz, tom 15 (2003), nomer 1.
Journal: St. Petersburg Math. J. 15 (2004), 49-79
MSC (2000): Primary 42B20, 42B25
Published electronically: December 31, 2003
MathSciNet review: 1979718
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Abstract: Coifman and Fefferman proved that the ``reverse Hölder inequality'' is fulfilled for any weight satisfying the Muckenhoupt condition. In order to illustrate the power of the Bellman function technique, Nazarov, Volberg, and Treil showed (among other things) how this technique leads to the reverse Hölder inequality for the weights satisfying the dyadic Muckenhoupt condition on the real line. In this paper the proof of the reverse Hölder inequality with sharp constants is presented for the weights satisfying the usual (rather than dyadic) Muckenhoupt condition on the line. The results are a consequence of the calculation of the true Bellman function for the corresponding extremal problem.


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Additional Information

V. Vasyunin
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191011, Russia
Email: vasyunin@pdmi.ras.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-03-00802-1
PII: S 1061-0022(03)00802-1
Keywords: Reverse H\"{o}lder inequality, Muckenhoupt weights, Bellman function
Received by editor(s): November 4, 2002
Published electronically: December 31, 2003
Additional Notes: Partially supported by RFBR (grant no. 02-01-00260a).
Article copyright: © Copyright 2003 American Mathematical Society