On Compactness of Commutators of Multiplications and Fourier Multipliers

Abstract

We generalise results on compactness of commutators of multiplications and Fourier multiplier operators by Cordes (J Funct Anal 18:115–131, 1975) with respect to the smoothness of multiplication function. Our prime motivation has been a particular case known as the first commutation lemma—the basic tool for defining H-measures and H-distributions. We review and improve the known results in the \(\mathrm{L}^p\) setting, with \(1<p<\infty \), illustrating these results with an application, obtaining a generalised compactness by compensation result.

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Correspondence to Nenad Antonić.

Additional information

This work was supported in part by the Croatian Science Foundation under project 9780 WeConMApp, by the bilateral Croatian–Montenegrin project Multiscale methods and calculus of variations, as well as by the project number 01-417 Advection–diffusion equations in highly heterogeneous media of the Montenegrin Ministry of Science. Part of this work was performed, while D. Mitrović was visiting University of Zagreb in the framework of the Marie Curie FP7-PEOPLE-2011-COFUND project Micro-local defect functionals and applications.

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Antonić, N., Mišur, M. & Mitrović, D. On Compactness of Commutators of Multiplications and Fourier Multipliers. Mediterr. J. Math. 15, 170 (2018). https://doi.org/10.1007/s00009-018-1215-8

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Mathematics Subject Classification

  • 42B15
  • 35B40
  • 28C15

Keywords

  • Commutator
  • compactness
  • Fourier multiplier
  • H-distribution