Maximal Weak-Type Inequality for Orthogonal Harmonic Functions and Martingales

Adam Osękowski. "Maximal Weak-Type Inequality for Orthogonal Harmonic Functions and Martingales." Bulletin of the Polish Academy of Sciences. Mathematics 61.3 (2013): 209-218. <http://eudml.org/doc/281235>.

@article{AdamOsękowski2013,
abstract = { 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. },
author = {Adam Osękowski},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {harmonic functions; martingales},
language = {eng},
number = {3},
pages = {209-218},
title = {Maximal Weak-Type Inequality for Orthogonal Harmonic Functions and Martingales},
url = {http://eudml.org/doc/281235},
volume = {61},
year = {2013},
}

TY - JOUR
AU - Adam Osękowski
TI - Maximal Weak-Type Inequality for Orthogonal Harmonic Functions and Martingales
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2013
VL - 61
IS - 3
SP - 209
EP - 218
AB - 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.
LA - eng
KW - harmonic functions; martingales
UR - http://eudml.org/doc/281235
ER -