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\(L^p\)-Boundedness and \(L^p\)-Nuclearity of Multilinear Pseudo-differential Operators on \({\mathbb {Z}}^n\) and the Torus \({\mathbb {T}}^n\)

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Abstract

In this article, we begin a systematic study of the boundedness and the nuclearity properties of multilinear periodic pseudo-differential operators and multilinear discrete pseudo-differential operators on \(L^p\)-spaces. First, we prove analogues of known multilinear Fourier multipliers theorems (proved by Coifman and Meyer, Grafakos, Tomita, Torres, Kenig, Stein, Fujita, Tao, etc.) in the context of periodic and discrete multilinear pseudo-differential operators. For this, we use the periodic analysis of pseudo-differential operators developed by Ruzhansky and Turunen. Later, we investigate the s-nuclearity, \(0<s \le 1,\) of periodic and discrete pseudo-differential operators. To accomplish this, we classify those s-nuclear multilinear integral operators on arbitrary Lebesgue spaces defined on \(\sigma \)-finite measures spaces. We also study similar properties for periodic Fourier integral operators. Finally, we present some applications of our study to deduce the periodic Kato–Ponce inequality and to examine the s-nuclearity of multilinear Bessel potentials as well as the s-nuclearity of periodic Fourier integral operators admitting suitable types of singularities.

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggestions which help us to improve the presentation of this article. Vishvesh Kumar thanks the Council of Scientific and Industrial Research, India, for its senior research fellowship. Duván Cardona was partially supported by the Department of Mathematics, Pontificia Universidad Javeriana.

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  1. Duván Cardona & Vishvesh Kumar

    Present address: Department of Mathematics, Analysis Logic and Discrete Mathematics, Ghent University, Ghent, Belgium

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  1. Department of Mathematics, Pontificia Universidad Javeriana, Bogotá, Colombia

    Duván Cardona

  2. Department of Mathematics, Indian Institute of Technology Delhi, Huaz Khas, New Delhi, 110016, India

    Vishvesh Kumar

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  1. Duván Cardona
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Correspondence to Duván Cardona.

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Communicated by Michael Ruzhansky.

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Cardona, D., Kumar, V. \(L^p\)-Boundedness and \(L^p\)-Nuclearity of Multilinear Pseudo-differential Operators on \({\mathbb {Z}}^n\) and the Torus \({\mathbb {T}}^n\). J Fourier Anal Appl 25, 2973–3017 (2019). https://doi.org/10.1007/s00041-019-09689-7

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