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 ℙ(|v|* ≥ 1)≤ (1 + 1/3² + 1/5² + 1/7² + ...)/(1 - 1/3² + 1/5² - 1/7² + ...) 𝔼u*. 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba61-3-3,
author = {Adam Os\k ekowski},
title = {Maximal Weak-Type Inequality for Orthogonal Harmonic Functions and Martingales},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {61},
year = {2013},
pages = {209-218},
zbl = {1302.31009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba61-3-3}
}
Adam Osękowski. Maximal Weak-Type Inequality for Orthogonal Harmonic Functions and Martingales. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 61 (2013) pp. 209-218. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba61-3-3/