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      One modification of the martingale transform and its applications to paraproducts and stochastic integrals

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          Abstract

          In this paper we introduce a variant of Burkholder's martingale transform associated with two martingales with respect to different filtrations. Even though the classical martingale techniques cannot be applied, we show that the discussed transformation still satisfies some expected \(\mathrm{L}^p\) estimates. Then we apply the obtained inequalities to general-dilation twisted paraproducts, particular instances of which have already appeared in the literature. As another application we construct stochastic integrals \(\int_{0}^{t}H_s d(X_s Y_s)\) associated with certain continuous-time martingales \((X_t)_{t\geq 0}\) and \((Y_t)_{t\geq 0}\). The process \((X_t Y_t)_{t\geq 0}\) is shown to be a "good integrator", although it is not necessarily a semimartingale, or even adapted to any convenient filtration.

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          Maximal and singular integral operators via Fourier transform estimates

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            Problems in harmonic analysis related to curvature

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              Strong variational and jump inequalities in harmonic analysis

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                Author and article information

                Journal
                2013-07-24
                2015-02-04
                Article
                10.1016/j.jmaa.2015.02.015
                1307.6403
                bdc1606e-aecd-4836-a8d2-f05ac4822496

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                Custom metadata
                Primary 60G42, Secondary 42B15, 60G46, 60H05
                J. Math. Anal. Appl. 426 (2015), 1143-1163
                18 pages; v3: accepted for publication, references updated, research support acknowledged
                math.PR

                Probability
                Probability

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