Sharp LlogL inequalities for differentially subordinated martingales and harmonic functions

Adam Oşekowski

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

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

|dx_n|≤|dy_n|, 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.

Primary Subjects: 60G42, 31B05

Full-text: Open access

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Permanent link to this document: http://projecteuclid.org/euclid.ijm/1254403712

Illinois Journal of Mathematics