On the best constant in the martingale version of Fefferman’s inequality

Abstract

Let $X=(X_{t})_{t\geq 0}\in H^{1}$ and $Y=(Y_{t})_{t\geq 0}\in{\mathrm{BMO}} $ be arbitrary continuous-path martingales. The paper contains the proof of the inequality \begin{equation*}\mathbb{E}\int _{0}^{\infty }\bigl\vert d\langle X,Y\rangle_{t}\bigr\vert \leq \sqrt{2}\Vert X\Vert _{H^{1}}\Vert Y\Vert _{\mathrm{BMO}_{2}},\end{equation*} and the constant $\sqrt{2}$ is shown to be the best possible. The proof rests on the construction of a certain special function, enjoying appropriate size and concavity conditions.