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Illinois Journal of Mathematics

Sharp LlogL inequalities for differentially subordinated martingales and harmonic functions

Adam Oşekowski
Source: Illinois J. Math. Volume 52, Number 3 (2008), 745-756.

Abstract

Let (xn), (yn) be two martingales adapted to the same filtration $(\mathcal{F}_{n})$ satisfying, with probability 1,

|dxn|≤|dyn|,  n=0, 1, 2, ….

For every K>0, we determine the best constant L=L(K) for which the inequality

\[\mathbb{E}|x_{n}|\leq K\mathbb{E}|y_{n}|\log|y_{n}|+L,\quad n=0,1,2,\ldots \]

holds true. We also prove a similar inequality for harmonic functions.

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Primary Subjects: 60G42, 31B05
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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.ijm/1254403712
Mathematical Reviews number (MathSciNet): MR2546005
Zentralblatt MATH identifier: 05660146

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