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Beyond Gevrey regularity

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Abstract

We define and study classes of smooth functions which are less regular than Gevrey functions. To that end we introduce two-parameter dependent sequences which do not satisfy Komatsu’s condition (M.2)’, which implies stability under differential operators within the spaces of ultradifferentiable functions. Our classes therefore have particular behavior under the action of differentiable operators. On a more advanced level, we study microlocal properties and prove that

$$\begin{aligned} {\text {WF}}_{0,\infty }(P(D)u)\subseteq {\text {WF}}_{0,\infty }(u)\subseteq {\text {WF}}_{0,\infty }(P(D)u) \cup \mathrm{Char}(P), \end{aligned}$$

where u is a Schwartz distribution, P(D) is a partial differential operator with constant coefficients and \({\text {WF}}_{0,\infty }\) is the wave front set described in terms of new regularity conditions. For the analysis we introduce particular admissibility condition for sequences of cut-off functions, and a new technical tool called enumeration.

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Acknowledgments

This research is supported by Ministry of Education, Science and Technological Development of Serbia through the Project No. 174024.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Novi Sad, Serbia

    Stevan Pilipović & Nenad Teofanov

  2. Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia

    Filip Tomić

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  1. Stevan Pilipović
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  2. Nenad Teofanov
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  3. Filip Tomić
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Correspondence to Nenad Teofanov.

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Pilipović, S., Teofanov, N. & Tomić, F. Beyond Gevrey regularity. J. Pseudo-Differ. Oper. Appl. 7, 113–140 (2016). https://doi.org/10.1007/s11868-016-0145-0

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  • DOI: https://doi.org/10.1007/s11868-016-0145-0

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