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Maximal Weak-Type Inequality for Orthogonal Harmonic Functions and Martingales

Osękowski A.

Bulletin of the Polish Academy of Sciences. Mathematics

2013

Vol. 61, no 3-4

209--218

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.

Weak-type inequality for the martingale square function and a related characterization of Hilbert spaces

Osękowski A.

Probability and Mathematical Statistics

2011

Vol. 31, Fasc. 2

227--238

Let f be a martingale taking values in a Banach space B and let S(f) be its square function. We show that if B is a Hilbert space, then P(S(f) ≥1)≤√e∥f∥1and the constant √e is the best possible. This extends the result of Cox, who established this bound in the real case. Next, we show that this inequality characterizes Hilbert spaces in the following sense: if B is not a Hilbert space, then there is a martingale f for which the above weak-type estimate does not hold.

A Weak-Type Inequality for Orthogonal Submartingales and Subharmonic Functions

Osękowski A.

Bulletin of the Polish Academy of Sciences. Mathematics

2011

Vol. 59, no 3

261--274

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.

Weak type inequality for the square function of a nonnegative submartingale

Osękowski A.

Bulletin of the Polish Academy of Sciences. Mathematics

2009

Vol. 57, no 1

81-89

Let ƒ be a nonnegative submartingale and S(ƒ) denote its square function. We show that for any λ > 0, λP(S(ƒ) ≥ λ) ≤ π/2||ƒ||1, and the constant π/2 is the best possible. The inequality is strict provided ||ƒ||1 ≠ 0.