## Electronic Communications in Probability

### A ratio inequality for nonnegative martingales and their differential subordinates

#### Abstract

We prove a sharp inequality for the ratio of the maximal functions of nonnegative martingales and their differential subordinates. An application in the theory of weighted inequalities is given.

#### Article information

Source
Electron. Commun. Probab., Volume 25 (2020), paper no. 21, 12 pp.

Dates
Accepted: 24 February 2020
First available in Project Euclid: 29 February 2020

https://projecteuclid.org/euclid.ecp/1582945214

Digital Object Identifier
doi:10.1214/20-ECP301

Mathematical Reviews number (MathSciNet)
MR4089728

Zentralblatt MATH identifier
07204043

#### Citation

Osękowski, Adam. A ratio inequality for nonnegative martingales and their differential subordinates. Electron. Commun. Probab. 25 (2020), paper no. 21, 12 pp. doi:10.1214/20-ECP301. https://projecteuclid.org/euclid.ecp/1582945214

#### References

• [1] Bañuelos, R. and Wang, G.: Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms. Duke Math. J. 80, (1995), 575–600.
• [2] Borichev, A., Janakiraman, P. and Volberg, A.: On Burkholder function for orthogonal martingales and zeros of Legendre polynomials. Amer. J. Math. 135, (2013), 207–236.
• [3] Burkholder, D. L.: Boundary value problems and sharp inequalities for martingale transforms. Ann. Probab. 12, (1984), 647–702.
• [4] Dellacherie, C. and Meyer, P.-A.: Probabilities and potential B: Theory of martingales. North Holland, Amsterdam, 1982.
• [5] Fefferman, R. and Pipher, J.: Multiparameter operators and sharp weighted inequalities. Amer. J. Math. 119, (1997), 337–369.
• [6] Fefferman, C. and Stein, E. M.: Some maximal inequalities. Amer. J. Math. 93, (1971), 107–115.
• [7] Geiss, S., Montgomery-Smith, S. and Saksman, E.: On singular integral and martingale transforms. Trans. Amer. Math. Soc. 362, (2010), 553–575.
• [8] Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Education, Inc., New Jersey, 2004.
• [9] Kazamaki, N.: Continuous exponential martingales and BMO. Lect. Notes in Math. 1579, Springer-Verlag, Berlin, Heidelberg, 1994.
• [10] Lerner, A. K., Ombrosi, S. and Pérez, C.: 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 p.
• [11] Lerner, A. K., Ombrosi, S. and Pérez, C.: Weak type estimates for singular integrals related to a dual problem of Muckenhoupt-Wheeden. J. Fourier Anal. Appl. 15, (2009), 394–403.
• [12] Lerner, A. K., Ombrosi, S. and Pérez, C.: $A_{1}$ bounds for Calderón-Zygmund operators related to a problem of Muckenhoupt and Wheeden. Math. Res. Lett. 16, (2009), 149–156.
• [13] Osękowski, A.: Sharp martingale and semimartingale inequalities. Monografie Matematyczne 72, Birkhäuser, 2012.
• [14] Osękowski, A.: A weighted weak-type bound for Haar multipliers. Royal Soc. Edinburgh Proc. A 148A, (2018), 643–658.
• [15] Reguera, M. C.: On Muckenhoupt-Wheeden conjecture. Adv. Math. 227, (2011), 1436–1450.
• [16] Suh, J.: A sharp weak type $(p,p)$ inequality $(p>2)$ for martingale transforms and other subordinate martingales. Trans. Amer. Math. Soc. 357, (2005), 1545–1564.
• [17] Wang, G.: Differential subordination and strong differential subordination for continuous time martingales and related sharp inequalities. Ann. Probab. 23, (1995), 522–551.