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WEIGHTED INEQUALITIES FOR MARTINGALE TRANSFORMS AND STOCHASTIC INTEGRALS

Part of: Stochastic processes Linear function spaces and their duals Harmonic analysis in several variables

Published online by Cambridge University Press:  20 February 2017

Adam Osȩkowski
Adam Osȩkowski*
Affiliation:
Department of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland email [email protected]
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Abstract

The paper is devoted to the study of Fefferman–Stein inequalities for stochastic integrals. If $X$ is a martingale, $Y$ is the stochastic integral, with respect to $X$, of some predictable process taking values in $[-1,1]$, then for any weight $W$ belonging to the class $A_{1}$ we have the estimates $\Vert Y_{\infty }\Vert _{L^{p}(W)}\leqslant 8pp^{\prime }[W]_{A_{1}}\Vert X_{\infty }\Vert _{L^{p}(W)},$$1<p<\infty ,$ and $\Vert Y_{\infty }\Vert _{L^{1,\infty }(W)}\leqslant c[W]_{A_{1}}(1+\log [W]_{A_{1}})\Vert X_{\infty }\Vert _{L^{1}(W)}.$ The proofs rest on the Bellman function method: the inequalities are deduced from the existence of certain special functions, enjoying appropriate majorization and concavity. As an application, related statements for Haar multipliers are indicated. The above estimates can be regarded as probabilistic counterparts of the recent results of Lerner, Ombrosi and Pérez concerning singular integral operators.

MSC classification

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 60G44: Martingales with continuous parameter
Type
Research Article
Information
Mathematika , Volume 63 , Issue 2 , 2017 , pp. 433 - 450
Copyright
Copyright © University College London 2017 

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