# Article

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Keywords:
martingale; stochastic integral; maximal inequality; differential subordination
Summary:
Assume that $X$, $Y$ are continuous-path martingales taking values in $\mathbb R^\nu$, $\nu \geq 1$, such that $Y$ is differentially subordinate to $X$. The paper contains the proof of the maximal inequality $$\|\sup _{t\geq 0} |Y_t| \|_1\leq 2\|\sup _{t\geq 0} |X_t| \|_1.$$ The constant $2$ is shown to be the best possible, even in the one-dimensional setting of stochastic integrals with respect to a standard Brownian motion. The proof uses Burkholder's method and rests on the construction of an appropriate special function.
References:
[1] Bañuelos, R., Méndez-Hernandez, P. J.: Space-time Brownian motion and the Beurling-Ahlfors transform. Indiana Univ. Math. J. 52 (2003), 981-990. DOI 10.1512/iumj.2003.52.2218 | MR 2001941 | Zbl 1080.60043
[2] Bañuelos, R., Wang, G.: Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms. Duke Math. J. 80 (1995), 575-600. DOI 10.1215/S0012-7094-95-08020-X | MR 1370109 | Zbl 0853.60040
[3] Burkholder, D. L.: Boundary value problems and sharp inequalities for martingale transforms. Ann. Probab. 12 (1984), 647-702. DOI 10.1214/aop/1176993220 | MR 0744226 | Zbl 0556.60021
[4] Burkholder, D. L.: A sharp and strict $L^p$-inequality for stochastic integrals. Ann. Probab. 15 (1987), 268-273. DOI 10.1214/aop/1176992268 | MR 0877602
[5] Burkholder, D. L.: Explorations in martingale theory and its applications. Calcul des Probabilités Ec. d'Été, Saint-Flour/Fr. 1989, Lect. Notes Math. 1464 1-66 (1991), Springer, Berlin. DOI 10.1007/BFb0085167 | MR 1108183 | Zbl 0771.60033
[6] Burkholder, D. L.: Sharp norm comparison of martingale maximal functions and stochastic integrals. Proceedings of the Norbert Wiener Centenary Congress, 1994 (East Lansing, MI, 1994) 343-358 Proc. Sympos. Appl. Math., 52, Amer. Math. Soc. Providence, RI (1997). MR 1440921 | Zbl 0899.60040
[7] Dellacherie, C., Meyer, P. A.: Probabilities and Potential. B: Theory of Martingales. Transl. from the French and prep. by J. P. Wilson North-Holland Mathematics Studies, Amsterdam (1982). MR 0745449 | Zbl 0494.60002
[8] Geiss, S., Montgomery-Smith, S., Saksman, E.: On singular integral and martingale transforms. Trans. Am. Math. Soc. 362 (2010), 553-575. DOI 10.1090/S0002-9947-09-04953-8 | MR 2551497 | Zbl 1196.60078
[9] Nazarov, F. L., Volberg, A.: Heat extension of the Beurling operator and estimates for its norm. Algebra i Analiz 15 (2003), 142-158 Russian translation in St. Petersburg Math. J. 15 (2004), 563-573. MR 2068982
[10] Osękowski, A.: Sharp LlogL inequality for differentially subordinated martingales. Illinois J. Math. 52 (2008), 745-756. MR 2546005
[11] Osękowski, A.: Sharp inequality for martingale maximal functions and stochastic integrals. Illinois J. Math. 54 (2010), 1133-1156. MR 2928348 | Zbl 1260.60081
[12] Osękowski, A.: Maximal inequalities for continuous martingales and their differential subordinates. Proc. Am. Math. Soc. 139 (2011), 721-734. DOI 10.1090/S0002-9939-2010-10539-7 | MR 2736351 | Zbl 1219.60044
[13] Osękowski, A.: Maximal inequalities for martingales and their differential subordinates. (to appear) in J. Theor. Probab. DOI:10.1007/s10959-012-0458-8. DOI 10.1007/s10959-012-0458-8
[14] Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. 3rd ed., Grundlehren der Mathematischen Wissenschaften 293 Springer, Berlin, 1999. MR 1725357 | Zbl 1087.60040
[15] Suh, Y.: A sharp weak type $(p,p)$ inequality $(p>2)$ for martingale transforms and other subordinate martingales. Trans. Am. Math. Soc. 357 (2005), 1545-1564. DOI 10.1090/S0002-9947-04-03563-9 | MR 2115376 | Zbl 1061.60042
[16] Wang, G.: Differential subordination and strong differential subordination for continuous-time martingales and related sharp inequalities. Ann. Probab. 23 (1995), 522-551. DOI 10.1214/aop/1176988278 | MR 1334160 | Zbl 0832.60055

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