## Proceedings of the Japan Academy, Series A, Mathematical Sciences

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

#### Abstract

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

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

Dates
First available in Project Euclid: 6 November 2012

https://projecteuclid.org/euclid.pja/1352210374

Digital Object Identifier
doi:10.3792/pjaa.88.139

Mathematical Reviews number (MathSciNet)
MR3000891

Zentralblatt MATH identifier
1266.60080

#### Citation

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. https://projecteuclid.org/euclid.pja/1352210374

#### References

• R. Bañuelos and K. Bogdan, Lévy processes and Fourier multipliers, J. Funct. Anal. 250 (2007), no. 1, 197–213.
• R. Bañuelos and G. Wang, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms, Duke Math. J. 80 (1995), no. 3, 575–600.
• K. Bichteler, Stochastic integration and $L^{p}$-theory of semimartingales, Ann. Probab. 9 (1981), no. 1, 49–89.
• D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), no. 3, 647–702.
• D. L. Burkholder, Sharp inequalities for martingales and stochastic integrals, Astérisque 157–158 (1988), 75–94.
• C. Dellacherie and P.-A. Meyer, Probabilities and potential. B, translated from the French by J. P. Wilson, North-Holland Mathematics Studies, 72, North-Holland, Amsterdam, 1982.
• S. Geiss, S. Montgomery-Smith and E. Saksman, On singular integral and martingale transforms, Trans. Amer. Math. Soc. 362 (2010), no. 2, 553–575.
• J. Marcinkiewicz, Quelques théoremes sur les séries orthogonales, Ann. Soc. Polon. Math. 16 (1937), 84–96.
• B. Maurey, Système de Haar, in Séminaire Maurey-Schwartz 1974–1975: Espaces L$^{p}$, applications radonifiantes et géométrie des espaces de Banach, Exp. Nos. I et II, 26 pp. (erratum, p. 1), Centre Math., École Polytech., Paris, 1975.
• A. Osękowski, Sharp LlogL inequalities for differentially subordinated martingales and harmonic functions, Illinois J. Math. 52 (2008), no. 3, 745–756.
• A. Osękowski, Sharp moment inequalities for differentially subordinated martingales, Studia Math. 201 (2010), no. 2, 103–131.
• A. Osękowski, On relaxing the assumption of differential subordination in some martingale inequalities, Electron. Commun. Probab. 16 (2011), 9–21.
• R. E. A. C. Paley, A Remarkable Series of Orthogonal Functions (I), Proc. London Math. Soc. 34 (1932), 241–264.
• D. Revuz and M. Yor, Continuous martingales and Brownian motion, third edition, Grundlehren der Mathematischen Wissenschaften, 293, Springer, Berlin, 1999.
• Y. Suh, A sharp weak type $(p,p)$ inequality $(p>2)$ for martingale transforms and other subordinate martingales, Trans. Amer. Math. Soc. 357 (2005), no. 4, 1545–1564 (electronic).
• G. Wang, Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities, Ann. Probab. 23 (1995), no. 2, 522–551.