Hostname: page-component-7857688df4-p2b9w Total loading time: 0 Render date: 2025-11-15T18:37:15.987Z Has data issue: false hasContentIssue false
  • English
  • Français
Get access Rights & Permissions [Opens in a new window]

Abstract

We show a result on propagation of the anisotropic Gabor wave front set for linear operators with a tempered distribution Schwartz kernel. The anisotropic Gabor wave front set is parametrized by a positive parameter relating the space and frequency variables. The anisotropic Gabor wave front set of the Schwartz kernel is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schrödinger equation for the free particle. The Laplacian is replaced by any partial differential operator with constant coefficients, real symbol and order at least two.

Keywords

tempered distributionsglobal wave front setsmicrolocal analysisphase spaceanisotropypropagation of singularitiesevolution equations

MSC classification

Primary: 46F05: Topological linear spaces of test functions, distributions and ultradistributions 46F12: Integral transforms in distribution spaces 35A27: Microlocal methods; methods of sheaf theory and homological algebra in PDE 47G30: Pseudodifferential operators 35S05: Pseudodifferential operators 35A18: Wave front sets 81S30: Phase-space methods including Wigner distributions, etc. 58J47: Propagation of singularities; initial value problems 47D06: One-parameter semigroups and linear evolution equations

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 3 , August 2024 , pp. 674 - 698
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Carypis, E. and Wahlberg, P., Propagation of exponential phase space singularities for Schrödinger equations with quadratic Hamiltonians, J. Fourier Anal. Appl. 23 (3) (2017), 530–571. Correction: 27(35) (2021).CrossRefGoogle Scholar
Cappiello, M., Gramchev, T. and Rodino, L., Entire extensions and exponential decay for semilinear elliptic equations, J. Anal. Math. 111 (2010), 339–367.CrossRefGoogle Scholar
Cappiello, M., Gramchev, T., Pilipovic, S. and Rodino, L., Anisotropic Shubin operators and eigenfunction expansions in Gelfand-Shilov spaces, J. Anal. Math. 138 (2019), 857–870.CrossRefGoogle Scholar
Cordero, E. and Rodino, L., Time-Frequency Analysis of Operators, De Gruyter Studies in Mathematics 75, De Gruyter, Berlin, 2020.CrossRefGoogle Scholar
Folland, G. B., Harmonic Analysis in Phase Space, Princeton University Press, 1989.CrossRefGoogle Scholar
Gröchenig, K., Foundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001.CrossRefGoogle Scholar
Hitrik, M. and Pravda–Starov, K., Spectra and semigroup smoothing for non-elliptic quadratic operators, Math. Ann. 344 (2009), 801–846.CrossRefGoogle Scholar
Hörmander, L., The Analysis of Linear Partial Differential Operators, I, III, IV, Springer, Berlin, 1990.Google Scholar
Hörmander, L., Quadratic hyperbolic operators, Microlocal Analysis and Applications, Lecture Notes in Math. 1495, L. Cattabriga, L. Rodino, 118–160, Springer, 1991.Google Scholar
Hörmander, L., Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z. 219 (1995), 413–449.CrossRefGoogle Scholar
Krantz, S. G. and Parks, H. R., The Implicit Function Theorem. History, Theory, and Applications, Birkhäuser, Boston, 2003.CrossRefGoogle Scholar
Nicola, F. and Rodino, L., Global Pseudo-Differential Calculus on Euclidean Spaces, Birkhäuser, Basel, 2010.CrossRefGoogle Scholar
Nakamura, S., Propagation of the homogeneous wave front set for Schrödinger equations, Duke Math. J. 126 (2) (2005), 349–367.CrossRefGoogle Scholar
Parenti, C. and Rodino, L., Parametrices for a class of pseudo differential operators I, Ann. Mat. Pura Appl. 125 (4) (1980), 221–254.CrossRefGoogle Scholar
Pravda-Starov, K., Rodino, L. and Wahlberg, P., Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians, Math. Nachr. 291 (1) (2018), 128–159.CrossRefGoogle Scholar
Reed, M. and Simon, B., Methods of Modern Mathematical Physics, I, Academic Press, 1980.Google Scholar
Rodino, L. and Wahlberg, P., The Gabor wave front set, Monaths. Math. 173 (4) (2014), 625–655.CrossRefGoogle Scholar
Rodino, L. and Wahlberg, P., Microlocal analysis of Gelfand–Shilov spaces, Ann. Mat. Pura Appl. 202 (2023), 2379–2420.CrossRefGoogle Scholar
Rodino, L. and Wahlberg, P., Anisotropic global microlocal analysis for tempered distributions, Monatsh. Math. 202 (2) (2023), 397–434.CrossRefGoogle Scholar
Schulz, R. and Wahlberg, P., Equality of the homogeneous and the Gabor wave front set, Comm. Partial Differential Equations 42 (5) (2017), 703–730.CrossRefGoogle Scholar
Shubin, M. A., Pseudodifferential Operators and Spectral Theory, Springer, 2001.CrossRefGoogle Scholar
Wahlberg, P., Propagation of polynomial phase space singularities for Schrödinger equations with quadratic Hamiltonians, Math. Scand. 122 (1) (2018), 107–140.CrossRefGoogle Scholar
Wahlberg, P., The Gabor wave front set of compactly supported distributions, Advances in Microlocal and Time-Frequency Analysis, P. Boggiatto, M. Cappiello, E. Cordero, S. Coriasco, G. Garello, A. Oliaro, J. Seiler (Eds.), Birkhäuser Verlag, 507–520, 2020.CrossRefGoogle Scholar
Wahlberg, P., Propagation of anisotropic Gelfand–Shilov wave front sets, J. Pseudo-Differ. Op. Appl. 14 (1) (2023).Google Scholar
Zhu, H., Propagation of singularities for gravity-capillary water waves, Analysis & PDE 17 (1) (2024), 281–344.CrossRefGoogle Scholar