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Inequalities for second-order Riesz transforms associated with Bessel expansions

100%

Osękowski A.

Bulletin of the Polish Academy of Sciences. Mathematics

2020

tom Vol. 68, no. 1

75--88

The paper contains the proofs of Lp, logarithmic and weak-type estimates for the second-order Riesz transforms arising in the context of multidimensional Bessel expansions. Using a novel probabilistic approach, which rests on martingale methods and the representation of Riesz transforms via associated Bessel-heat processes, we show that these estimates hold with constants independent of the dimension.

Weighted Maximal Inequalities for Martingale Transforms

63%

Brzozowski M. , Osękowski A.

Probability and Mathematical Statistics

2021

tom 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.

Doob's estimate for coherent random variables and maximal operators on trees

63%

Cichomski S. , Osękowski A.

Probability and Mathematical Statistics

2023

tom Vol. 43, Fasc. 1

109--119

Let ξ be an integrable random variable defined on (Ω, F, P). Fix k ∈ Z+ and let {Gji }1≤i≤n,1≤j≤k be a reference family of sub-σ-fields of F such that {Gji }1≤i≤n is a filtration for each j ∈ {1, . . . , k}. In this article we explain the underlying connection between the analysis of the maximal functions of the corresponding coherent vector and basic combinatorial properties of the uncentered Hardy-Littlewood maximal operator. Following a classical approach of Grafakos, Kinnunen and Montgomery-Smith, we establish an appropriate version of Doob’s celebrated maximal estimate.