On the best constant in the estimate related to 𝐻¹-𝐵𝑀𝑂 duality

Osękowski, Adam

doi:10.1090/proc/15234

On the best constant in the estimate related to 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 , $\varphi$ be arbitrary elements of and , 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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