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Second commutation lemma for fractional H-measures

  • Marko Erceg
  • Ivan Ivec
Article

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.

Keywords

H-measures Fractional derivatives Propagation principle 

Mathematics Subject Classification

35R11 35S05 46G10 47B47 47G30 

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Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceUniversity of ZagrebZagrebCroatia
  2. 2.Scuola Internazionale Superiore di Studi AvanzatiTriesteItaly
  3. 3.Faculty of MetallurgyUniversity of ZagrebSisakCroatia

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