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

Let I ⊂ R I\subset \mathbb {R} be an interval and let f f , φ \varphi be arbitrary elements of H 1 ( I ) H^1(I) and B M O ( I ) BMO(I) , respectively, with ∫ I φ = 0 \int _I\varphi =0 . The paper contains the proof of the estimate ∫ I f φ ≤ 2 ‖ f ‖ H 1 ( I ) ‖ φ ‖ B M O ( I ) , \begin{equation*} \int _I f\varphi \leq \sqrt {2}\|f\|_{H^1(I)}\|\varphi \|_{BMO(I)}, \end{equation*} and it is shown that 2 \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.