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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 41, Fasc. 1,
pages 89 - 114
DOI: 10.37190/0208-4147.41.1.7
Published online 1.4.2021
 

Weighted maximal inequalities for martingale~transforms

Michał Brzozowski
Adam Osękowski

Abstract: We study the weighted maximal \(L^{1}\)-inequality for martingale transforms, under the assumption that the underlying weight satisfies Muckenhoupt’s condition \(A_\infty\) and that the filtration is regular. The resulting linear dependence of the constant on the \(A_\infty\) characteristic of the weight is optimal. The proof exploits certain special functions enjoying appropriate size conditions and concavity.

2010 AMS Mathematics Subject Classification: Primary 60G44; Secondary 60G42.

Keywords and phrases: martingale, weight, Bellman function, maximal function.

D. L. Burkholder, Boundary value problems and sharp inequalities for martingale transforms, Ann. Probab. 12 (1984), 647-702.

A. Osękowski, Weighted weak-type inequality for martingales, Bull. Polish Acad. Sci. Math. 65 (2017), 165-175.

A. Osękowski, Weighted maximal inequalities for the Haar system, Monatsh. Math. 186 (2018), 321-336.

A. Osękowski, Weighted square function inequalities, Publ. Mat. 62 (2018), 321-336.

Y. Suh, A sharp weak type (p, p) inequality (p > 2) for martingale transforms and other subordinate martingales, Trans. Amer. Math. Soc. 357 (2005), 1545-1564

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