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Second commutation lemma for fractional H-measures

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Abstract

Classical H-measures introduced by Tartar (Proc R Soc Edinb 115A:193–230, 1990) and independently by Gérard (Commun Partial Differ Equ 16:1761–1794, 1991) are essentially suited for hyperbolic equations while parabolic equations fit in the framework of the parabolic H-measures developed by Antonić and Lazar (2007–2013). More recently the study of differential relations with fractional derivatives prompted the extension of the theory to arbitrary ratios, thus the fractional H-measures were introduced and applied to fractional conservation laws by Mitrović and Ivec (Commun Pure Appl Anal 10(6):1617–1627, 2011). In this paper we explore the transport property of fractional H-measures by studying fractional derivatives of commutators of multiplication and Fourier multiplier operators. In particular, we prove the Second commutation lemma suitable for fractional H-measures, comprehending the known hyperbolic and parabolic cases, while allowing for derivation of the corresponding propagation principle for fractional H-measures. At the end, on a model example we present this derivation of the transport equation for the fractional H-measure.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, Zagreb, Croatia

    Marko Erceg

  2. Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea 265, Trieste, Italy

    Marko Erceg

  3. Faculty of Metallurgy, University of Zagreb, Aleja narodnih heroja 3, Sisak, Croatia

    Ivan Ivec

Authors
  1. Marko Erceg
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  2. Ivan Ivec
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Correspondence to Ivan Ivec.

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Dedicated to the memory of professor Todor V. Gramchev.

This work has been supported in part by Croatian Science Foundation under the project 9780 WeConMApp, and by University of Zagreb trough Grant PMF-M02/2016, as well as by the bilateral Croatian–Serbian project Microlocal analysis, partial differential equations and application to heterogeneous materials. The first author has partially been supported by 2014–2017 MIUR-FIR Grant No. RBFR13WAET.

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Erceg, M., Ivec, I. Second commutation lemma for fractional H-measures. J. Pseudo-Differ. Oper. Appl. 9, 589–613 (2018). https://doi.org/10.1007/s11868-017-0207-y

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  • DOI: https://doi.org/10.1007/s11868-017-0207-y

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