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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

Micha³ Brzozowski
Adam Osêkowski

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.

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