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2 Bulletin of the Polish Academy of Sciences. Mathematics

2 Osękowski A.

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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.