Let M, N be real-valued martingales such that N is differentially subordinate to M. The paper contains the proofs of the following weak-type inequalities:
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(i) If M≥0 and 0p, ∞≤2‖M‖p
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and the constant is the best possible.
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(ii) If M≥0 and p≥2, then
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\[\Vert N\Vert_{p,\infty}\leq\frac{p}{2}(p-1)^{-1/p}\Vert M\Vert_{p}\]
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and the constant is the best possible.
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(iii) If 1≤p≤2 and M and N are orthogonal, then
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‖N‖p, ∞≤Kp‖M‖p
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where
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\[K_{p}^{p}=\frac{1}{\Gamma(p+1)}\cdot\biggl(\frac{\pi}{2}\biggr)^{p-1}\cdot\frac{1+1/3^{2}+1/5^{2}+1/7^{2}+\cdots}{1-1/3^{p+1}+1/5^{p+1}-1/7^{p+1}+\cdots}.\]
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The constant is the best possible.
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We also provide related estimates for harmonic functions on Euclidean domains.
Publié le : 2009-08-15
Classification:
differential subordination,
harmonic function,
martingale
@article{1251463285,
author = {Osekowski, Adam},
title = {Sharp weak-type inequalities for differentially subordinated martingales},
journal = {Bernoulli},
volume = {15},
number = {1},
year = {2009},
pages = { 871-897},
language = {en},
url = {http://dml.mathdoc.fr/item/1251463285}
}
Osȩkowski, Adam. Sharp weak-type inequalities for differentially subordinated martingales. Bernoulli, Tome 15 (2009) no. 1, pp. 871-897. http://gdmltest.u-ga.fr/item/1251463285/