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Schatten class and nuclear pseudo-differential operators on homogeneous spaces of compact groups

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Abstract

Given a compact (Hausdorff) group G and a closed subgroup H of G,  in this paper we present symbolic criteria for pseudo-differential operators on the compact homogeneous space G/H characterizing the Schatten–von Neumann classes \(S_r(L^2(G/H))\) for all \(0<r \le \infty .\) We go on to provide a symbolic characterization for r-nuclear, \(0< r \le 1,\)  pseudo-differential operators on \(L^{p}(G/H)\) with applications to adjoint, product and trace formulae. The criteria here are given in terms of matrix-valued symbols defined on noncommutative analogue of phase space \(G/H \times \widehat{G/H}.\) Finally, we present an application of aforementioned results in the context of the heat kernels.

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Acknowledgements

The authors thank the referee for several suggestions leading to the improvement of the manuscript. Vishvesh Kumar is supported by the FWO Odysseus 1 Grant G.0H94.18N: Analysis and Partial Differential Equations and Partial Differential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant No. 01M01021). The authors thank Prof. Michael Ruzhansky for providing many valuable comments. Vishvesh Kumar thanks Duván Cardona for many fruitful discussions. Shyam Swarup Mondal thanks the Council of Scientific and Industrial Research, India, for providing financial support. He also thanks his supervisor Jitendriya Swain for his support and encouragement.

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Authors and Affiliations

  1. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, Building S8, 9000, Ghent, Belgium

    Vishvesh Kumar

  2. Department of Mathematics, IIT Guwahati, Guwahati, Assam, India

    Shyam Swarup Mondal

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  1. Vishvesh Kumar
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  2. Shyam Swarup Mondal
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Kumar, V., Mondal, S.S. Schatten class and nuclear pseudo-differential operators on homogeneous spaces of compact groups. Monatsh Math 197, 149–176 (2022). https://doi.org/10.1007/s00605-021-01663-0

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