Doob’s and Burkholder-Davis-Gundy inequalities with variable exponent

Weisz, Ferenc

doi:10.1090/proc/15262

Doob’s and Burkholder-Davis-Gundy inequalities with variable exponent
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Proc. Amer. Math. Soc. 149 (2021), 875-888 Request permission

Abstract:

Let $p(\cdot )$ be a measurable function defined on a probability space $\Omega$ and $p_- \colonequals \inf _{x\in \Omega }p(x)$, $p_+\colonequals \sup _{x\in \Omega }p(x)$. Under a probabilistic version of the log-Hölder continuity of $1/p(\cdot )$, Doob’s inequality is proved if $1<p_- \leq p_+ \leq \infty$. Dual Doob’s inequality, the Davis decomposition and the generalization of the Burkholder-Davis-Gundy inequality is also verified for $1 \leq p_- \leq p_+<\infty$.

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