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On the best constant in the estimate related to $ H^1-BMO$ duality


Author: Adam Osękowski
Journal: Proc. Amer. Math. Soc. 149 (2021), 333-343
MSC (2010): Primary 42A05, 42B35, 49K20
DOI: https://doi.org/10.1090/proc/15234
Published electronically: October 20, 2020
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Abstract: Let $ I\subset \mathbb{R}$ be an interval and let $ f$, $ \varphi $ be arbitrary elements of $ H^1(I)$ and $ BMO(I)$, respectively, with $ \int _I\varphi =0$. The paper contains the proof of the estimate

$\displaystyle \int _I f\varphi \leq \sqrt {2}\Vert f\Vert _{H^1(I)}\Vert\varphi \Vert _{BMO(I)},$    

and it is shown that $ \sqrt {2}$ cannot be replaced by a smaller universal constant. The argument rests on the existence of a special function enjoying appropriate size and concavity requirements.

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Additional Information

Adam Osękowski
Affiliation: Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
Email: A.Osekowski@mimuw.edu.pl

DOI: https://doi.org/10.1090/proc/15234
Keywords: BMO, Hardy space, Bellman function
Received by editor(s): January 15, 2020
Received by editor(s) in revised form: June 12, 2020
Published electronically: October 20, 2020
Additional Notes: This research was supported by Narodowe Centrum Nauki (Poland), grant no. DEC-2014/ 14/E/ST1/00532.
Communicated by: Ariel Barton
Article copyright: © Copyright 2020 American Mathematical Society