Skip to main content Accessibility help


  • Adam Osȩkowski (a1)


The paper is devoted to the study of Fefferman–Stein inequalities for stochastic integrals. If $X$ is a martingale, $Y$ is the stochastic integral, with respect to $X$ , of some predictable process taking values in $[-1,1]$ , then for any weight $W$ belonging to the class $A_{1}$ we have the estimates $\Vert Y_{\infty }\Vert _{L^{p}(W)}\leqslant 8pp^{\prime }[W]_{A_{1}}\Vert X_{\infty }\Vert _{L^{p}(W)},$ $1<p<\infty ,$ and $\Vert Y_{\infty }\Vert _{L^{1,\infty }(W)}\leqslant c[W]_{A_{1}}(1+\log [W]_{A_{1}})\Vert X_{\infty }\Vert _{L^{1}(W)}.$ The proofs rest on the Bellman function method: the inequalities are deduced from the existence of certain special functions, enjoying appropriate majorization and concavity. As an application, related statements for Haar multipliers are indicated. The above estimates can be regarded as probabilistic counterparts of the recent results of Lerner, Ombrosi and Pérez concerning singular integral operators.



Hide All
1. Buckley, S. M., Estimates for operator norms on weighted spaces and reverse Jensen inequalities. Trans. Amer. Math. Soc. 340(1) 1993, 253272.
2. Burkholder, D. L., Boundary value problems and sharp inequalities for martingale transforms. Ann. Probab. 12 1984, 647702.
3. Dellacherie, C. and Meyer, P.-A., Probabilities and Potential B: Theory of Martingales, North-Holland (Amsterdam, 1982).
4. Fefferman, C. and Stein, E. M., Some maximal inequalities. Amer. J. Math. 93 1971, 107115.
5. Izumisawa, M. and Kazamaki, N., Weighted norm inequalities for martingales. Tohoku Math. J. (2) 29 1977, 115124.
6. Kazamaki, N., Continuous Exponential Martingales and BMO (Lecture Notes in Mathematics 1579 ), Springer (Berlin, 1994).
7. 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.
8. 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, 149156.
9. Nazarov, F. L., Reznikov, A., Vasyunin, V. and Volberg, A., A Bellman function counterexample to the $A_{1}$ conjecture: the blow-up of the weak norm estimates of weighted singular operators. Preprint, 2015, arXiv:1506.04710.
10. Nazarov, F. L. and Treil, S. R., The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis. St. Petersburg Math. J. 8 1997, 721824.
11. Nazarov, F. L., Treil, S. R. and Volberg, A., The Bellman functions and two-weight inequalities for Haar multipliers. J. Amer. Math. Soc. 12 1999, 909928.
12. Obłój, J., The Skorokhod embedding problem and its offspring. Probab. Surv. 1 2004, 321392.
13. Osȩkowski, A., Sharp Martingale and Semimartingale Inequalities (Monografie Matematyczne 72 ), Birkhäuser (Basel, 2012), 462 pp.
14. Petermichl, S. and Wittwer, J., A sharp estimate for the weighted Hilbert transform via Bellman functions. Michigan Math. J. 50 2002, 7187.
15. Reguera, M. C., On Muckenhoupt–Wheeden conjecture. Adv. Math. 227(4) 2011, 14361450.
16. Reguera, M. C. and Thiele, C., The Hilbert transform does not map L 1(Mw) to L 1, (w). Math. Res. Lett. 19(1) 2012, 17.
17. Slavin, L. and Vasyunin, V., Sharp results in the integral-form John–Nirenberg inequality. Trans. Amer. Math. Soc. 363 2011, 41354169.
18. Vasyunin, V. and Volberg, A., Monge–Ampére equation and Bellman optimization of Carleson Embedding Theorems. In Linear and Complex Analysis (American Mathematical Society Translations (2) 226 ), American Mathematical Society (Providence, RI, 2009), 195238.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed