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3 Bulletin of the Polish Academy of Sciences. Mathematics

3 Osękowski A.

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Sharp Logarithmic Inequalities for Two Hardy-type Operators

Osękowski A.

Bulletin of the Polish Academy of Sciences. Mathematics

2015

Vol. 63, no. 3

237--247

For any locally integrable f on Rn, we consider the operators S and T which average f over balls of radius |x| and center 0 and x, respectively: [WZÓR] for x ∈ Rn. The purpose of the paper is to establish sharp localized LlogL estimates for S and T. The proof rests on a corresponding one-weight estimate for a martingale maximal function, a result which is of independent interest.

A Weak-Type Inequality for Submartingales and Itô Processes

Osękowski A.

Bulletin of the Polish Academy of Sciences. Mathematics

2015

Vol. 63, no. 1

73--88

Let α∈[0,1] be a fixed parameter. We show that for any nonnegative submartingale X and any semimartingale Y which is α-subordinate to X, we have the sharp estimate [WZÓR]. Here W is the weak-L∞space introduced by Bennett, DeVore and Sharpley. The inequality is already sharp in the context of α-subordinate Itô processes.

Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales

Osękowski A.

Bulletin of the Polish Academy of Sciences. Mathematics

2014

Vol. 62, no 2

187--196

Let (hk)k≥0 be the Haar system on [0,1]. We show that for any vectors ak from a separable Hilbert space H and any εk∈[−1,1], k=0,1,2,…, we have the sharp inequality ...[formula], where W([0,1]) is the weak-L∞ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound ∥Y∥W(Ω)≤2∥X∥L∞(Ω), where X and Y stand for H-valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.