Open Access
August 2020 On the best constant in the martingale version of Fefferman’s inequality
Adam Osękowski
Bernoulli 26(3): 1912-1926 (August 2020). DOI: 10.3150/19-BEJ1175


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.


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Adam Osękowski. "On the best constant in the martingale version of Fefferman’s inequality." Bernoulli 26 (3) 1912 - 1926, August 2020.


Received: 1 July 2019; Revised: 1 November 2019; Published: August 2020
First available in Project Euclid: 27 April 2020

zbMATH: 07193947
MathSciNet: MR4091096
Digital Object Identifier: 10.3150/19-BEJ1175

Keywords: $\mathrm{BMO}$ , best constants , Duality , martingale , maximal

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

Vol.26 • No. 3 • August 2020
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