Burkholder’s function and a weighted 𝐿² bound for stochastic integrals

Bañuelos, Rodrigo; Brzozowski, Michał; Osȩkowski, Adam

doi:10.1090/proc/15136

Burkholder’s function and a weighted $L^2$ bound for stochastic integrals
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by Rodrigo Bañuelos, Michał Brzozowski and Adam Osȩkowski

Proc. Amer. Math. Soc. 148 (2020), 5013-5028

DOI: https://doi.org/10.1090/proc/15136

Published electronically: August 4, 2020

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Abstract:

Let $X$ be a continuous-path martingale and let $Y$ be a stochastic integral, with respect to $X$, of some predictable process with values in $[-1,1]$. We provide an explicit formula for Burkholder’s function associated with the weighted $L^2$ bound \begin{equation*} \|Y\|_{L^2(W)}\lesssim [w]_{A_2}\|X\|_{L^2(W)}. \end{equation*}

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