Beyond Gevrey regularity

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• 作者：Stevan Pilipović ; Nenad Teofanov…
• 关键词：Ultradifferentiable functions ; Gevrey classes ; Ultradistributions ; Wave ; front sets
• 刊名：Journal of Pseudo-Differential Operators and Applications
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：7
• 期：1
• 页码：113-140
• 全文大小：643 KB
• 参考文献：1.Cappiello, M., Schulz, R.: Microlocal analysis of quasianalytic Gelfand-Shilov type ultradistributions, preprint versoin arXiv:​1309.​4236v1 [math.AP]. Accepted for publication in Complex Variables and Elliptic Equations
  2.Carypis, E., Wahlberg, P.: Propagation of exponential phase space singularities for Schrdinger equations with quadratic Hamiltonians, arXiv:​1510.​0032 [math.AP] (2015)
  3.Chen, H., Rodino, L.: General theory of PDE and Gevrey classes in General theory of partial differential equations and microlocal analysis. Pitman Res. Notes Math. Ser. 349, 6–81 (1996)MathSciNet
  4.Cordero, E., Nicola, F., Rodino, L.: Schrödinger equations with rough Hamiltonians. Discrete Contin. Dyn. Syst. 35(10), 4805–4821 (2015)CrossRef MathSciNet
  5.Cordero, E., Nicola, F., Rodino, L.: Propagation of the Gabor wave front set for Schrodinger equations with non-smooth potentials. Rev. Math. Phys. 27(1), 33 (2015)CrossRef MathSciNet
  6.Coriasco, S., Johansson, K., Toft, J.: Local wave-front sets of Banach and Frchet types, and pseudo-differential operators. Monatsh. Math. 169(3–4), 285–316 (2013)CrossRef MathSciNet MATH
  7.Coriasco, S., Johansson, K., Toft, J.: Global Wave-front Sets of Banach, Frchet and Modulation Space Types, and Pseudo-differential Operators. J. Differ. Equ. 245(8), 3228–3258 (2013)CrossRef MathSciNet
  8.Feichtinger, H.G.: Modulation spaces on locally compact abelian groups, Technical Report, University Vienna, 1983. Wavelets and Their Applications, pp. 99–140. Allied Publishers (2003)
  9.Feichtinger, H.G., Gröchenig, K.: Gabor frames and time-frequency analysis of distributions. J. Funct. Anal. 146, 464–495 (1997)CrossRef MathSciNet MATH
  10.Feichtinger, H.G., Strohmer, T. (eds.): Gabor Analysis and Algorithms: Theory and Applications. Birkhäuser, Boston (1998)MATH
  11.Feichtinger, H.G., Strohmer, T. (eds.): Advances in Gabor Analysis. Birkhäuser, Boston (2003)MATH
  12.Foland, G.B.: Harmonic Analysis in Phase Space. Princeton Univ, Press, Princeton (1989)
  13.Gevrey, M.: Sur la nature analitique des solutions des équations aux dérivées partielle. Ann. Ec. Norm. Sup. Paris 35, 129–190 (1918)MathSciNet MATH
  14.Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)CrossRef MATH
  15.Hörmander, L.: The Analysis of Linear Partial Differential Operators. Vol. I: Distribution Theory and Fourier Analysis. Springer-Verlag, Berlin (1983)
  16.Hörmander, L.: Quadratic hyperbolic operators. In: Microlocal Analysis and Applications (Montecatini Terme, 1989), Lecture Notes in Math. 1495, pp. 118–160. Springer, New York (1991)
  17.Johansson, K., Pilipović, S., Teofanov, N.: Discrete Wave-front sets of Fourier Lebesgue and modulation space types. Monatshefte fur Mathematik 166(2), 181–199 (2012)CrossRef MathSciNet MATH
  18.Klotz, A.: Inverse closed ultradifferential subalgebras. J. Math. Anal. Appl. 409(2), 615–629 (2014)CrossRef MathSciNet MATH
  19.Komatsu, H.: Ultradistributions, I: structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 20(1), 25–105 (1973)MathSciNet MATH
  20.Komatsu, H.: An introduction to the theory of generalized functions. Lecturenotes. Department of Mathematics Science University of Tokyo (1999)
  21.Narasimhan, R.: Analysis on Real and Complex Manifolds. North-Holland Mathematical Library, vol. 35. North-Holland Publishing Co., Amsterdam (1985)
  22.Pilipovic, S., Teofanov, N., Toft, J.: Micro-local analysis in Fourier Lebesgue and modulation spaces. Part I. J. Fourier Anal. Appl. 17(3), 374–407 (2011)CrossRef MathSciNet MATH
  23.Pilipovic, S., Teofanov, N., Toft, J.: Micro-local analysis in Fourier Lebesgue and modulation spaces, Part II. J Pseudo-Differ. Oper. Appl. 1(3), 341–376 (2010)CrossRef MathSciNet MATH
  24.Pilipović, S., Teofanov, N., Tomić, F.: On a class of ultradifferentiable functions. Novi Sad J. Math. 45(1), 125–142 (2015)MathSciNet
  25.Pilipović, S., Toft, J.: Wave-front sets related to quasi-analytic Gevrey sequences (2015). Preprint availible online at arXiv:​1210.​7741v3
  26.Pravda-Starov, K., Rodino, L., Wahlberg, P.: Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians (2015). arXiv:​1411.​0251v5 [math.AP]
  27.Rauch, J.: Partial Differential Equations. Springer-Verlag, Berlin (1991)CrossRef MATH
  28.Rodino, L.: Linear Partial Differential Operators in Gevrey Spaces. World Scientific, Singapore (1993)CrossRef MATH
  29.Rodino, L., Wahlberg, P.: The Gabor wave front set. Monatsh. Math. 173(4), 625–655 (2014)CrossRef MathSciNet MATH
  30.Schulz, R., Wahlberg, P.: The equality of the homogeneous and the Gabor wave front set (2013). arXiv:​1304.​7608v2 [math.AP]
  31.Schulz, R., Wahlberg, P.: Microlocal properties of Shubin pseudodifferential and localization operators. J. Pseudo-Differ. Oper. Appl. (2015). doi:10.​1007/​s11868-015-0143-7
  32.Wahlberg, P.: Propagation of polynomial phase space singularities for Schrdinger equations with quadratic Hamiltonians (2015). arXiv:​1411.​6518v3 [math.AP]
  
• 作者单位：Stevan Pilipović (1)
  Nenad Teofanov (1)
  Filip Tomić (2)
  
  1. Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Novi Sad, Serbia
  2. Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：None Assigned
• 出版者：Birkh盲user Basel
• ISSN：1662-999X

文摘

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. Keywords Ultradifferentiable functions Gevrey classes Ultradistributions Wave-front sets Mathematics Subject Classification Primary 35A18 46F05 Secondary 46F10 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (32) References1.Cappiello, M., Schulz, R.: Microlocal analysis of quasianalytic Gelfand-Shilov type ultradistributions, preprint versoin arXiv:​1309.​4236v1 [math.AP]. Accepted for publication in Complex Variables and Elliptic Equations2.Carypis, E., Wahlberg, P.: Propagation of exponential phase space singularities for Schrdinger equations with quadratic Hamiltonians, arXiv:​1510.​0032 [math.AP] (2015)3.Chen, H., Rodino, L.: General theory of PDE and Gevrey classes in General theory of partial differential equations and microlocal analysis. Pitman Res. Notes Math. Ser. 349, 6–81 (1996)MathSciNet4.Cordero, E., Nicola, F., Rodino, L.: Schrödinger equations with rough Hamiltonians. Discrete Contin. Dyn. Syst. 35(10), 4805–4821 (2015)CrossRefMathSciNet5.Cordero, E., Nicola, F., Rodino, L.: Propagation of the Gabor wave front set for Schrodinger equations with non-smooth potentials. Rev. Math. Phys. 27(1), 33 (2015)CrossRefMathSciNet6.Coriasco, S., Johansson, K., Toft, J.: Local wave-front sets of Banach and Frchet types, and pseudo-differential operators. Monatsh. Math. 169(3–4), 285–316 (2013)CrossRefMathSciNetMATH7.Coriasco, S., Johansson, K., Toft, J.: Global Wave-front Sets of Banach, Frchet and Modulation Space Types, and Pseudo-differential Operators. J. Differ. Equ. 245(8), 3228–3258 (2013)CrossRefMathSciNet8.Feichtinger, H.G.: Modulation spaces on locally compact abelian groups, Technical Report, University Vienna, 1983. Wavelets and Their Applications, pp. 99–140. Allied Publishers (2003)9.Feichtinger, H.G., Gröchenig, K.: Gabor frames and time-frequency analysis of distributions. J. Funct. Anal. 146, 464–495 (1997)CrossRefMathSciNetMATH10.Feichtinger, H.G., Strohmer, T. (eds.): Gabor Analysis and Algorithms: Theory and Applications. Birkhäuser, Boston (1998)MATH11.Feichtinger, H.G., Strohmer, T. (eds.): Advances in Gabor Analysis. Birkhäuser, Boston (2003)MATH12.Foland, G.B.: Harmonic Analysis in Phase Space. Princeton Univ, Press, Princeton (1989)13.Gevrey, M.: Sur la nature analitique des solutions des équations aux dérivées partielle. Ann. Ec. Norm. Sup. Paris 35, 129–190 (1918)MathSciNetMATH14.Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)CrossRefMATH15.Hörmander, L.: The Analysis of Linear Partial Differential Operators. Vol. I: Distribution Theory and Fourier Analysis. Springer-Verlag, Berlin (1983)16.Hörmander, L.: Quadratic hyperbolic operators. In: Microlocal Analysis and Applications (Montecatini Terme, 1989), Lecture Notes in Math. 1495, pp. 118–160. Springer, New York (1991)17.Johansson, K., Pilipović, S., Teofanov, N.: Discrete Wave-front sets of Fourier Lebesgue and modulation space types. Monatshefte fur Mathematik 166(2), 181–199 (2012)CrossRefMathSciNetMATH18.Klotz, A.: Inverse closed ultradifferential subalgebras. J. Math. Anal. Appl. 409(2), 615–629 (2014)CrossRefMathSciNetMATH19.Komatsu, H.: Ultradistributions, I: structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 20(1), 25–105 (1973)MathSciNetMATH20.Komatsu, H.: An introduction to the theory of generalized functions. Lecturenotes. Department of Mathematics Science University of Tokyo (1999)21.Narasimhan, R.: Analysis on Real and Complex Manifolds. North-Holland Mathematical Library, vol. 35. North-Holland Publishing Co., Amsterdam (1985)22.Pilipovic, S., Teofanov, N., Toft, J.: Micro-local analysis in Fourier Lebesgue and modulation spaces. Part I. J. Fourier Anal. Appl. 17(3), 374–407 (2011)CrossRefMathSciNetMATH23.Pilipovic, S., Teofanov, N., Toft, J.: Micro-local analysis in Fourier Lebesgue and modulation spaces, Part II. J Pseudo-Differ. Oper. Appl. 1(3), 341–376 (2010)CrossRefMathSciNetMATH24.Pilipović, S., Teofanov, N., Tomić, F.: On a class of ultradifferentiable functions. Novi Sad J. Math. 45(1), 125–142 (2015)MathSciNet25.Pilipović, S., Toft, J.: Wave-front sets related to quasi-analytic Gevrey sequences (2015). Preprint availible online at arXiv:​1210.​7741v3 26.Pravda-Starov, K., Rodino, L., Wahlberg, P.: Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians (2015). arXiv:​1411.​0251v5 [math.AP]27.Rauch, J.: Partial Differential Equations. Springer-Verlag, Berlin (1991)CrossRefMATH28.Rodino, L.: Linear Partial Differential Operators in Gevrey Spaces. World Scientific, Singapore (1993)CrossRefMATH29.Rodino, L., Wahlberg, P.: The Gabor wave front set. Monatsh. Math. 173(4), 625–655 (2014)CrossRefMathSciNetMATH30.Schulz, R., Wahlberg, P.: The equality of the homogeneous and the Gabor wave front set (2013). arXiv:​1304.​7608v2 [math.AP]31.Schulz, R., Wahlberg, P.: Microlocal properties of Shubin pseudodifferential and localization operators. J. Pseudo-Differ. Oper. Appl. (2015). doi:10.​1007/​s11868-015-0143-7 32.Wahlberg, P.: Propagation of polynomial phase space singularities for Schrdinger equations with quadratic Hamiltonians (2015). arXiv:​1411.​6518v3 [math.AP] About this Article Title Beyond Gevrey regularity Journal Journal of Pseudo-Differential Operators and Applications Volume 7, Issue 1 , pp 113-140 Cover Date2016-03 DOI 10.1007/s11868-016-0145-0 Print ISSN 1662-9981 Online ISSN 1662-999X Publisher Springer International Publishing Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Analysis Operator Theory Partial Differential Equations Functional Analysis Applications of Mathematics Algebra Keywords Ultradifferentiable functions Gevrey classes Ultradistributions Wave-front sets Primary 35A18 46F05 Secondary 46F10 Authors Stevan Pilipović (1) Nenad Teofanov (1) Filip Tomić (2) Author Affiliations 1. Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Novi Sad, Serbia 2. Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia Continue reading... To view the rest of this content please follow the download PDF link above.