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1

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

Cichomski Stanisław, Osękowski Adam

Probability and Mathematical Statistics

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2023

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Vol. 43, Fasc. 1

109--119

EN

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.

2

Sharp inequalities for the Haar system and martingale transforms

Osękowski A.

Probability and Mathematical Statistics

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2016

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Vol. 36, Fasc. 1

147--164

EN

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.

3

Moment Inequality for the Martingale Square Function

Osękowski A.

Bulletin of the Polish Academy of Sciences. Mathematics

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2013

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Vol. 61, no 2

169--180

EN

Consider the sequence (Cn)n≥1 of positive numbers defined by C1=1 and Cn+1=1+C2n/4, n=1,2,…. Let M be a real-valued martingale and let S(M) denote its square function. We establish the bound E|Mn|≤CnESn(M), n=1,2,…, and show that for each n, the constant Cn is the best possible.

4

Maximal Weak-Type Inequality for Orthogonal Harmonic Functions and Martingales

Osękowski A.

Bulletin of the Polish Academy of Sciences. Mathematics

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2013

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Vol. 61, no 3-4

209--218

EN

Assume that u, v are conjugate harmonic functions on the unit disc of C, normalized so that u(0)=v(0)=0. Let u∗, |v|∗ stand for the one- and two-sided Brownian maxima of u and v, respectively. The paper contains the proof of the sharp weak-type estimate... [formula]. Actually, this estimate is shown to be true in the more general setting of differentially subordinate harmonic functions defined on Euclidean domains. The proof exploits a novel estimate for orthogonal martingales satisfying differential subordination.

5

A Weak-Type Inequality for Orthogonal Submartingales and Subharmonic Functions

Osękowski A.

Bulletin of the Polish Academy of Sciences. Mathematics

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2011

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Vol. 59, no 3

261--274

EN

Let X be a submartingale starting from 0, and Y be a semimartingale which is orthogonal and strongly differentially subordinate to X. The paper contains the proof of the sharp estimate P(supt≥0|Yt|≥1)≤3.375…∥X∥1. As an application, a related weak-type inequality for smooth functions on Euclidean domains is established.