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Local bounds for singular Brascamp–Lieb forms with cubical structure

Mathematische Zeitschrift volume 302, pages 2375–2405 (2022)Cite this article

Abstract

We prove a range of \(L^p\) bounds for singular Brascamp–Lieb forms with cubical structure. We pass through sparse and local bounds, the latter proved by an iteration of Fourier expansion, telescoping, and the Cauchy–Schwarz inequality. We allow \(2^{m-1}<p\le \infty \) with m the dimension of the cube, extending an earlier result that required \(p=2^m\). The threshold \(2^{m-1}\) is sharp in our theorems.

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Acknowledgements

The first author is supported by NSF DMS-2154356. The second author is supported by the Primus research programme PRIMUS/21/SCI/002 of Charles University. The third author acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy- EXC-2047/1-390685813 as well as SFB 1060. Part of the research was carried out during a delightful workshop on Real and Harmonic Analysis at the Oberwolfach Research Institute for Mathematics. The authors thank the anonymous referee for their careful reading of the paper and valuable suggestions.

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Authors and Affiliations

  1. Schmid College of Science and Technology, Chapman University, One University Drive, Orange, CA, 92866, USA

    Polona Durcik

  2. Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Lenka Slavíková

  3. Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Christoph Thiele

Authors
  1. Polona Durcik
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  2. Lenka Slavíková
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  3. Christoph Thiele
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Correspondence to Polona Durcik.

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Durcik, P., Slavíková, L. & Thiele, C. Local bounds for singular Brascamp–Lieb forms with cubical structure. Math. Z. 302, 2375–2405 (2022). https://doi.org/10.1007/s00209-022-03148-8

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  • DOI: https://doi.org/10.1007/s00209-022-03148-8

Keywords

  • Multilinear form
  • Singular integral
  • Sparse bounds

Mathematics Subject Classification

  • 42B20