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 $H^1-BMO$ duality
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by Adam Osękowski

Proc. Amer. Math. Soc. 149 (2021), 333-343

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 \begin{equation*} \int _I f\varphi \leq \sqrt {2}\|f\|_{H^1(I)}\|\varphi \|_{BMO(I)}, \end{equation*} 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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