Sharp weak-type inequalities for differentially subordinated martingales

Abstract

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: (i) If $M\ge 0$ and $0<p\le 1$, then \[\Vert N\Vert_{p,\infty}\leq2\Vert M\Vert_p\] and the constant is the best possible. (ii) If $M\ge 0$ and $p\ge 2$, then \[\Vert N\Vert_{p,\infty}\leq\frac{p}{2}(p-1)^{-1/p}\Vert M\Vert_p\] and the constant is the best possible. (iii) If $1\le p\le 2$ and $M$ and $N$ are orthogonal, then \[\Vert N\Vert_{p,\infty}\leq K_p\Vert M\Vert_p,\] where \[K_p^p=\frac{1}{Γ(p+1)}\cdot\biggl(\fracπ{2}\biggr)^{p-1}\cdot\frac{1+1/3^2+1/5^2+1/7^2+...}{1-1/3^{p+1}+1/5^ {p+1}-1/7^{p+1}+...}.\] The constant is the best possible. We also provide related estimates for harmonic functions on Euclidean domains.

Published in at http://dx.doi.org/10.3150/08-BEJ166 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)