Proceedings of the Japan Academy, Series A, Mathematical Sciences

Weak $\Phi$-inequalities for the Haar system and differentially subordinated martingales

Adam Osȩkowski

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For a wide class of Young functions $\Phi:[0,\infty)\to[0,\infty)$, we determine the best constant $C_{\Phi}$ such that the following holds. If $(h_{k})_{k\geq 0}$ is the Haar system on $[0,1]$, then for any vectors $a_{k}$ from a separable Hilbert space $\mathcal{H}$ and $\varepsilon_{k}\in \{-1,1\}$, $k=0, 1, 2,\ldots$, we have \begin{equation*} \left|\left\{x\in [0,1]:\left|∑_{k=0}^{n} ɛ_{k}a_{k}h_{k}(x)\right|≥ 1\right\}\right|≤ C_{Φ} ∫_{0}^{1}Φ\left(\left|∑_{k=0}^{n} a_{k}h_{k}(x)\right|\right)\mathrm{d}x,\quad n=0,1,2,…. \end{equation*} This is generalized to the sharp weak-$\Phi$ inequality \begin{equation*} \mathbf{P}(\sup_{t≥ 0}|Y_{t}|≥ 1)≤ C_{Φ}\sup_{t≥ 0}\mathbf{E} Φ(|X_{t}|), \end{equation*} where $X$, $Y$ stand for $\mathcal{H}$-valued martingales such that $Y$ is differentially subordinate to $X$. These statements complement and generalize the results of Burkholder, Suh, the author and others.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 88, Number 9 (2012), 139-144.

First available in Project Euclid: 6 November 2012

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Zentralblatt MATH identifier

Primary: 60G44: Martingales with continuous parameter
Secondary: 60G42: Martingales with discrete parameter 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Haar system martingale weak-$\Phi$ inequality best constant


Osȩkowski, Adam. Weak $\Phi$-inequalities for the Haar system and differentially subordinated martingales. Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 9, 139--144. doi:10.3792/pjaa.88.139.

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