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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Sharp weak type inequalities for the Haar system and related estimates for nonsymmetric martingale transforms


Author: Adam Osȩkowski
Journal: Proc. Amer. Math. Soc. 140 (2012), 2513-2526
MSC (2010): Primary 60G42; Secondary 60G44
Published electronically: November 8, 2011
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Abstract: For any $ 1\leq p<\infty $, we determine the optimal constant $ C_p$ such that the following holds. If $ (h_k)_{k\geq 0}$ is the Haar system, then for any vectors $ a_k$ from a separable Hilbert space $ \mathcal {H}$ and $ \theta _k\in \{0,1\}$, $ k=0,\,1,\,2,\,\ldots $, we have

$\displaystyle \left \vert\left \vert\sum _{k=0}^n \theta _ka_kh_k\right \vert\r... ...}\leq C_p\left \vert\left \vert\sum _{k=0}^n a_kh_k\right \vert\right \vert _p.$

This is generalized to the weak type inequality

$\displaystyle \vert\vert g\vert\vert _{p,\infty }\leq C_p\vert\vert f\vert\vert _p,$

where $ f$ is an $ \mathcal {H}$-valued martingale and $ g$ is its transform by a predictable sequence taking values in $ [0,1]$. We extend this further to the estimate

$\displaystyle \vert\vert Y\vert\vert _{p,\infty }\leq C_p\vert\vert X\vert\vert _p,$

valid for any two $ \mathcal {H}$-valued continuous-time martingales $ X$, $ Y$, such that $ ([Y,X-Y]_t)$ is nondecreasing and nonnegative as a function of $ t$.

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Additional Information

Adam Osȩkowski
Affiliation: Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
Email: ados@mimuw.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11093-1
PII: S 0002-9939(2011)11093-1
Keywords: Martingale, martingale transform, differential subordination, weak type inequality, unconditional constant, Haar system.
Received by editor(s): July 8, 2010
Received by editor(s) in revised form: February 21, 2011
Published electronically: November 8, 2011
Additional Notes: Partially supported by MNiSW Grant N N201 397437.
Communicated by: Richard C. Bradley
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.