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A Decomposition of Calderón–Zygmund Type and Some Observations on Differentiation of Integrals on the Infinite-Dimensional Torus

Abstract

In this note we will show a Calderón–Zygmund decomposition associated with a function \(f\in L^{1}(\mathbb {T}^{\omega })\). The idea relies on an adaptation of a more general result by J. L. Rubio de Francia in the setting of locally compact groups. Some related results about differentiation of integrals on the infinite-dimensional torus are also discussed.

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Acknowledgments

The authors would like to thank the referees for their very careful reading and useful comments which indeed improved the presentation of the paper.

Author information

Correspondence to Luz Roncal.

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The second author is supported by the Basque Government through the BERC 2018-2021 program, by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2017-2018 and through project MTM2017-82160-C2-1-P funded by (AEI/FEDER, UE) and acronym “HAQMEC”, and by a2017 Leonardo grant for Researchers and Cultural Creators, BBVA Foundation. The Foundation accepts no responsibility for the opinions, statements and contents included in the project and/or the results thereof, which are entirely the responsibility of the authors.

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Fernández, E., Roncal, L. A Decomposition of Calderón–Zygmund Type and Some Observations on Differentiation of Integrals on the Infinite-Dimensional Torus. Potential Anal (2020). https://doi.org/10.1007/s11118-019-09813-8

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Keywords

  • Infinite dimensional torus
  • Calderón–Zygmund decomposition
  • Differentiation of integrals
  • Differentiation basis
  • Locally compact groups

Mathematics Subject Classification (2010)

  • Primary: 42B05
  • Secondary: 20E07, 43A70