Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A weighted maximal inequality for differentially subordinate martingales


Authors: Rodrigo Bañuelos and Adam Osękowski
Journal: Proc. Amer. Math. Soc. 146 (2018), 2263-2275
MSC (2010): Primary 60G44; Secondary 42B25
DOI: https://doi.org/10.1090/proc/13912
Published electronically: January 12, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The paper contains the proof of a weighted Fefferman-Stein inequality in a probabilistic setting. Suppose that $ f=(f_n)_{n\geq 0}$, $ g=(g_n)_{n\geq 0}$ are martingales such that $ g$ is differentially subordinate to $ f$, and let $ w=(w_n)_{n\geq 0}$ be a weight, i.e., a nonnegative, uniformly integrable martingale. Denoting by $ Mf=\sup _{n\geq 0}\vert f_n\vert$, $ Mw=\sup _{n\geq 0}w_n$ the maximal functions of $ f$ and $ w$, we prove the weighted inequality

$\displaystyle \vert\vert g\vert\vert _{L^1(w)}\leq C\vert\vert Mf\vert\vert _{L^1(Mw)},$    

where $ C=3+\sqrt {2}+4\ln 2=7.186802\ldots $. The proof rests on the existence of a special function enjoying appropriate majorization and concavity.

References [Enhancements On Off] (What's this?)

  • [1] Rodrigo Bañuelos and Gang Wang, Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms, Duke Math. J. 80 (1995), no. 3, 575-600. MR 1370109, https://doi.org/10.1215/S0012-7094-95-08020-X
  • [2] D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), no. 3, 647-702. MR 744226
  • [3] Donald L. Burkholder, Explorations in martingale theory and its applications, École d'Été de Probabilités de Saint-Flour XIX--1989, Lecture Notes in Math., vol. 1464, Springer, Berlin, 1991, pp. 1-66. MR 1108183, https://doi.org/10.1007/BFb0085167
  • [4] Donald L. Burkholder, Sharp norm comparison of martingale maximal functions and stochastic integrals, Proceedings of the Norbert Wiener Centenary Congress, 1994 (East Lansing, MI, 1994) Proc. Sympos. Appl. Math., vol. 52, Amer. Math. Soc., Providence, RI, 1997, pp. 343-358. MR 1440921, https://doi.org/10.1090/psapm/052/1440921
  • [5] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. MR 0284802, https://doi.org/10.2307/2373450
  • [6] Tuomas Hytönen, On Petermichl's dyadic shift and the Hilbert transform, C. R. Math. Acad. Sci. Paris 346 (2008), no. 21-22, 1133-1136 (English, with English and French summaries). MR 2464252, https://doi.org/10.1016/j.crma.2008.09.021
  • [7] Tuomas Hytönen, Carlos Pérez, Sergei Treil, and Alexander Volberg, Sharp weighted estimates for dyadic shifts and the $ A_2$ conjecture, J. Reine Angew. Math. 687 (2014), 43-86. MR 3176607, https://doi.org/10.1515/crelle-2012-0047
  • [8] Andrei K. Lerner, Sheldy Ombrosi, and Carlos Pérez, Sharp $ A_1$ bounds for Calderón-Zygmund operators and the relationship with a problem of Muckenhoupt and Wheeden, Int. Math. Res. Not. IMRN 6 (2008), Art. ID rnm161, 11. MR 2427454, https://doi.org/10.1093/imrn/rnm161
  • [9] Andrei K. Lerner, Sheldy Ombrosi, and Carlos Pérez, $ A_1$ bounds for Calderón-Zygmund operators related to a problem of Muckenhoupt and Wheeden, Math. Res. Lett. 16 (2009), no. 1, 149-156. MR 2480568, https://doi.org/10.4310/MRL.2009.v16.n1.a14
  • [10] F. L. Nazarov, A. Reznikov, V. Vasyunin, and A. Volberg, Weak norm estimates of weighted singular operators and Bellman functions. Manuscript (2010).
  • [11] F. L. Nazarov and S. R. Treĭl, The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis, Algebra i Analiz 8 (1996), no. 5, 32-162 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 5, 721-824. MR 1428988
  • [12] F. Nazarov, S. Treil, and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), no. 4, 909-928. MR 1685781, https://doi.org/10.1090/S0894-0347-99-00310-0
  • [13] F. Nazarov, S. Treil, and A. Volberg, Bellman function in stochastic control and harmonic analysis, Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000) Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 393-423. MR 1882704
  • [14] Adam Osekowski, Sharp martingale and semimartingale inequalities, Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], vol. 72, Birkhäuser/Springer Basel AG, Basel, 2012. MR 2964297
  • [15] Adam Oseowski, Maximal inequalities for continuous martingales and their differential subordinates, Proc. Amer. Math. Soc. 139 (2011), no. 2, 721-734. MR 2736351, https://doi.org/10.1090/S0002-9939-2010-10539-7
  • [16] Adam Osekowski, Maximal inequalities for martingales and their differential subordinates, J. Theoret. Probab. 27 (2014), no. 1, 1-21. MR 3174213, https://doi.org/10.1007/s10959-012-0458-8
  • [17] Stefanie Petermichl, Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol, C. R. Acad. Sci. Paris Sér. I Math. 330 (2000), no. 6, 455-460 (English, with English and French summaries). MR 1756958, https://doi.org/10.1016/S0764-4442(00)00162-2
  • [18] Maria Carmen Reguera, On Muckenhoupt-Wheeden conjecture, Adv. Math. 227 (2011), no. 4, 1436-1450. MR 2799801, https://doi.org/10.1016/j.aim.2011.03.009
  • [19] Maria Carmen Reguera and Christoph Thiele, The Hilbert transform does not map $ L^1(Mw)$ to $ L^{1,\infty}(w)$, Math. Res. Lett. 19 (2012), no. 1, 1-7. MR 2923171, https://doi.org/10.4310/MRL.2012.v19.n1.a1
  • [20] L. Slavin and V. Vasyunin, Sharp results in the integral-form John-Nirenberg inequality, Trans. Amer. Math. Soc. 363 (2011), no. 8, 4135-4169. MR 2792983, https://doi.org/10.1090/S0002-9947-2011-05112-3
  • [21] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • [22] Vasily Vasyunin and Alexander Volberg, Monge-Ampère equation and Bellman optimization of Carleson embedding theorems, Linear and complex analysis, Amer. Math. Soc. Transl. Ser. 2, vol. 226, Amer. Math. Soc., Providence, RI, 2009, pp. 195-238. MR 2500520, https://doi.org/10.1090/trans2/226/16

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 60G44, 42B25

Retrieve articles in all journals with MSC (2010): 60G44, 42B25


Additional Information

Rodrigo Bañuelos
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: banuelos@math.purdue.edu

Adam Osękowski
Affiliation: Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
Email: ados@mimuw.edu.pl

DOI: https://doi.org/10.1090/proc/13912
Keywords: Martingale, differential subordination, weight
Received by editor(s): January 4, 2017
Received by editor(s) in revised form: July 30, 2017
Published electronically: January 12, 2018
Additional Notes: The first author was supported in part by NSF Grant #0603701-DMS
The second author was supported in part by the NCN grant DEC-2014/14/E/ST1/00532.
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society