Algebraic Calderón-Zygmund theory

Junge, M., Mei, T., Parcet, J. and Xia, R. (2021) Algebraic Calderón-Zygmund theory. Advances in Mathematics, 376, 107443. (doi: 10.1016/j.aim.2020.107443)

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Calderón-Zygmund theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of good metrics, we introduce a new approach for general measure spaces which admit a Markov semigroup satisfying purely algebraic assumptions. We shall construct an abstract form of ‘Markov metric’ governing the Markov process and the naturally associated BMO spaces, which interpolate with the Lp-scale and admit endpoint inequalities for Calderón-Zygmund operators. Motivated by noncommutative harmonic analysis, this approach gives the first form of Calderón-Zygmund theory for arbitrary von Neumann algebras, but is also valid in classical settings like Riemannian manifolds with nonnegative Ricci curvature or doubling/nondoubling spaces. Other less standard commutative scenarios like fractals or abstract probability spaces are also included. Among our applications in the noncommutative setting, we improve recent results for quantum Euclidean spaces and group von Neumann algebras, respectively linked to noncommutative geometry and geometric group theory.

Item Type:Articles
Additional Information:Mei is partially supported by NSF grant DMS1700171. Javier Parcet is supported by the Europa Excelencia Grant MTM2016-81700-ERC and the CSIC Grant PIE-201650E030. Javier Parcet and Runlian Xia are supported by ICMAT Severo Ochoa Grant SEV-2015-0554 (Spain) and “Ayuda extraordinaria a Centros de Excelencia Severo Ochoa” (20205CEX001).
Glasgow Author(s) Enlighten ID:Xia, Dr Runlian
Authors: Junge, M., Mei, T., Parcet, J., and Xia, R.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Advances in Mathematics
ISSN (Online):1090-2082
Published Online:19 November 2020

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