Burkholder–Davis–Gundy Inequalities in UMD Banach Spaces,Communications in Mathematical Physics

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Burkholder–Davis–Gundy Inequalities in UMD Banach Spaces
Communications in Mathematical Physics ( IF 2.2 ) Pub Date : 2020-09-16 , DOI: 10.1007/s00220-020-03845-7
Ivan Yaroslavtsev

In this paper we prove Burkholder-Davis-Gundy inequalities for a general martingale $M$ with values in a UMD Banach space $X$. Assuming that $M_0=0$, we show that the following two-sided inequality holds for all $1\leq p<\infty$: \begin{align}\label{eq:main}\tag{$\star$} \mathbb E \sup_{0\leq s\leq t} \|M_s\|^p \eqsim_{p, X} \mathbb E \gamma([\![M]\!]_t)^p ,\;\;\; t\geq 0. \end{align} Here $ \gamma([\![M]\!]_t) $ is the $L^2$-norm of the unique Gaussian measure on $X$ having $[\![M]\!]_t(x^*,y^*):= [\langle M,x^*\rangle, \langle M,y^*\rangle]_t$ as its covariance bilinear form. This extends to general UMD spaces a recent result by Veraar and the author, where a pointwise version of \eqref{eq:main} was proved for UMD Banach functions spaces $X$. We show that for continuous martingales, \eqref{eq:main} holds for all $0

中文翻译：

UMD Banach 空间中的 Burkholder-Davis-Gundy 不等式

在本文中，我们证明了在 UMD Banach 空间 $X$ 中具有值的一般鞅$M$ 的 Burkholder-Davis-Gundy 不等式。假设 $M_0=0$，我们证明以下两侧不等式对所有 $1\leq p<\infty$ 成立： \begin{align}\label{eq:main}\tag{$\star$} \ mathbb E \sup_{0\leq s\leq t} \|M_s\|^p \eqsim_{p, X} \mathbb E \gamma([\![M]\!]_t)^p ,\;\ ;\; t\geq 0. \end{align} 这里 $ \gamma([\![M]\!]_t) $ 是在 $X$ 上具有 $[\! [M]\!]_t(x^*,y^*):= [\langle M,x^*\rangle, \langle M,y^*\rangle]_t$ 作为其协方差双线性形式。这扩展到了一般 UMD 空间，这是 Veraar 和作者最近的结果，其中为 UMD Banach 函数空间 $X$ 证明了 \eqref{eq:main} 的逐点版本。我们证明，对于连续的鞅，\eqref{eq:main} 对所有 $0 都成立

更新日期：2020-09-16

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