Bernoulli

  • Bernoulli
  • Volume 15, Number 3 (2009), 871-897.

Sharp weak-type inequalities for differentially subordinated martingales

Adam Osȩkowski

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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≥0 and 0<p≤1, then

Np, ∞≤2‖Mp

and the constant is the best possible.

(ii) If M≥0 and p≥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≤p≤2 and M and N are orthogonal, then

Np, ∞KpMp

where

\[K_{p}^{p}=\frac{1}{\Gamma(p+1)}\cdot\biggl(\frac{\pi}{2}\biggr)^{p-1}\cdot\frac{1+1/3^{2}+1/5^{2}+1/7^{2}+\cdots}{1-1/3^{p+1}+1/5^{p+1}-1/7^{p+1}+\cdots}.\]

The constant is the best possible.

We also provide related estimates for harmonic functions on Euclidean domains.

Article information

Source
Bernoulli, Volume 15, Number 3 (2009), 871-897.

Dates
First available in Project Euclid: 28 August 2009

Permanent link to this document
https://projecteuclid.org/euclid.bj/1251463285

Digital Object Identifier
doi:10.3150/08-BEJ166

Mathematical Reviews number (MathSciNet)
MR2555203

Keywords
differential subordination harmonic function martingale

Citation

Osȩkowski, Adam. Sharp weak-type inequalities for differentially subordinated martingales. Bernoulli 15 (2009), no. 3, 871--897. doi:10.3150/08-BEJ166. https://projecteuclid.org/euclid.bj/1251463285


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