Multilinear singular integrals on non-commutative $$L^p$$ L p spaces
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Mathematische Annalen
Multilinear singular integrals on non-commutative Lp spaces Francesco Di Plinio1
· Kangwei Li2,3 · Henri Martikainen4 · Emil Vuorinen5
Received: 14 August 2019 / Revised: 21 July 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We prove L p bounds for the extensions of standard multilinear Calderón–Zygmund operators to tuples of UMD spaces tied by a natural product structure. The product can, for instance, mean the pointwise product in UMD function lattices, or the composition of operators in the Schatten-von Neumann subclass of the algebra of bounded operators on a Hilbert space. We do not require additional assumptions beyond UMD on each space—in contrast to previous results, we e.g. show that the Rademacher maximal function property is not necessary. The obtained generality allows for novel applications. For instance, we prove new versions of fractional Leibniz rules via our results concerning the boundedness of multilinear singular integrals in non-commutative L p spaces. Our proof techniques combine a novel scheme of induction on the multilinearity index with dyadic-probabilistic techniques in the UMD space setting. Keywords Calderón–Zygmund operators · Singular integrals · Multilinear analysis · Non-commutative spaces · Representation theorems · UMD spaces Mathematics Subject Classification 42B20 (primary) · 46E40 · 46L52 (secondary)
Communicated by Loukas Grafakos. F. Di Plinio has been partially supported by the National Science Foundation under the grants NSF-DMS-1650810, NSF-DMS-1800628 and NSF-DMS-2000510. K. Li was supported by Juan de la Cierva—Formación 2015 FJCI-2015-24547, by the Basque Government through the BERC 2018–2021 program and by Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and through project MTM2017-82160-C2-1-P funded by (AEI/FEDER, UE) and acronym “HAQMEC”. H. Martikainen was supported by the Academy of Finland through the Grants 294840 and 306901, and by the 3-year research Grant 75160010 of the University of Helsinki. He is a member of the Finnish Centre of Excellence in Analysis and Dynamics Research. E. Vuorinen was supported by the Academy of Finland through the Grant 306901, by the Finnish Centre of Excellence in Analysis and Dynamics Research, and by Jenny and Antti Wihuri Foundation. Extended author information available on the last page of the article
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1 Introduction A Banach space X has the UMD property if any X -valued martingale difference sequence converges unconditionally in L p for some (equivalently, all) p ∈ (1, ∞). Standard examples of UMD spaces are provided by the reflexive L p function spaces, as well as the reflexive Schatten–von Neumann subclasses S p of the algebra of bounded operators on a Hilbert space. The works by Burkholder [2] and Bourgain [1] yield an alternative characterization: X is a UMD space if and only if singular integrals, in particular the Hilbert transform, admit an L p (X )-bounded extension.
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