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Weighted Maximal Inequalities for Martingale Transforms

Brzozowski Michał, Osękowski Adam

Probability and Mathematical Statistics

2021

Vol. 41, Fasc. 1

89--114

We study the weighted maximal L1-inequality for martingale transforms, under the assumption that the underlying weight satisfies Muckenhoupt’s condition A∞ and that the filtration is regular. The resulting linear dependence of the constant on the A∞ characteristic of the weight is optimal. The proof exploits certain special functions enjoying appropriate size conditions and concavity.

Bellman functions and L^p estimates for paraproducts

Kovač V., Škreb K. A.

Probability and Mathematical Statistics

2018

Vol. 38, Fasc. 2

459--479

We give an explicit formula for one possible Bellman function associated with the Lp boundedness of dyadic paraproducts regarded as bilinear operators or trilinear forms. Then we apply the same Bellman function in various other settings, to give self-contained alternative proofs of the estimates for several classical operators. These include the martingale paraproducts of Bañuelos and Bennett and the paraproducts with respect to the heat flows.

Sharp inequalities for the Haar system and martingale transforms

Osękowski A.

Probability and Mathematical Statistics

2016

Vol. 36, Fasc. 1

147--164

A classical result of Paley and Marcinkiewicz asserts that the Haar system on [0, 1] forms an unconditional basis in Lp provided 1 < p < ∞. The purpose of the paper is to study related weak-type inequalities, which can be regarded as a version of this property for p = 1. Probabilistic counterparts, leading to some sharp estimates for martingale transforms, are presented.