Propagation of anisotropic Gelfand-Shilov wave front sets. (English) Zbl 1522.46031
The author considers the analogue of the Gabor wave-front set in the setting of anisotropic Gelfand-Shilov ultradistributions of Beurling type \(\Sigma^{s\,'}_t(\mathbb{R}^d)\), \(s+t>1\). It is defined as follows. Let \(u\in\Sigma^{s\,'}_t(\mathbb{R}^d)\) and let \(\psi\) be a test function in \(\Sigma^s_t(\mathbb{R}^d)\). Then \((x_0,\xi_0)\in\mathbb{R}^d\times (\mathbb{R}^d\backslash\{0\})\) does not belong to the anisotropic \(t,s\)-Gelfand-Shilov wave front set \(WF^{t,s}(u)\) of \(u\) if there is an open set \(U\subseteq \mathbb{R}^d\times (\mathbb{R}^d\backslash\{0\})\) containing \((x_0,\xi_0)\) such that \[ \sup_{\lambda>0,\, (x,\xi)\in U} e^{r\lambda}|V_{\psi}u(\lambda^tx,\lambda^s\xi)|<\infty\text{ for all }r>0; \] here, \(V_{\psi}u\) stands for the short-time Fourier transform of \(u\) with window \(\psi\). The main result of the article describes the propagation of singularities with respect to this wave front set of an operator \(\mathcal{K}:\Sigma^{s\,'}_t(\mathbb{R}^d)\rightarrow \Sigma^{s\,'}_t(\mathbb{R}^d)\); i.e., it shows that for \(u\in\Sigma^{s\,'}_t(\mathbb{R}^d)\), \(WF^{t,s}(\mathcal{K}u)\) is included in a set built, in a precise manner, out of \(WF^{t,s}(K)\) and \(WF^{t,s}(u)\) where \(K\) is the kernel of \(\mathcal{K}\).
In the last section, the author applies this result to a class of evolution equations.
In the last section, the author applies this result to a class of evolution equations.
Reviewer: Bojan Prangoski (Skopje)
MSC:
46F05 | Topological linear spaces of test functions, distributions and ultradistributions |
46F12 | Integral transforms in distribution spaces |
35A27 | Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs |
47G30 | Pseudodifferential operators |
35A18 | Wave front sets in context of PDEs |
81S30 | Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics |
58J47 | Propagation of singularities; initial value problems on manifolds |
47D06 | One-parameter semigroups and linear evolution equations |
Keywords:
Gelfand-Shilov spaces; ultradistributions; global wave front sets; microlocal analysis; phase space; anisotropy; propagation of singularities; evolution equationsReferences:
[1] | Abdeljawad, A.; Cappiello, M.; Toft, J., Pseudo-differential calculus in anisotropic Gelfand-Shilov setting, Integr. Equ. Oper. Theory., 91, 26 (2019) · Zbl 1419.35262 · doi:10.1007/s00020-019-2518-2 |
[2] | Boggiatto, P.; Oliaro, A.; Wahlberg, P., The wave front set of the Wigner distribution and instantaneous frequency, J. Fourier Anal. Appl., 18, 410-438 (2012) · Zbl 1248.42011 · doi:10.1007/s00041-011-9201-6 |
[3] | Cappiello, M.; Schulz, R., Microlocal analysis of quasianalytic Gelfand-Shilov type ultradistributions, Compl. Var. Elliptic Equ., 61, 4, 538-561 (2016) · Zbl 1339.35014 · doi:10.1080/17476933.2015.1106481 |
[4] | Carypis, E., Wahlberg, P.: Propagation of exponential phase space singularities for Schrödinger equations with quadratic Hamiltonians. J. Fourier Anal. Appl. 23(3), 530-571 (2017). (Correction: 27:35 (2021)) · Zbl 1377.35228 |
[5] | Cordero, E.; Rodino, L., Time-Frequency Analysis of Operators (2020), Berlin: De Gruyter, Berlin · Zbl 07204958 · doi:10.1515/9783110532456 |
[6] | Gel’fand, IM; Shilov, GE, Generalized Functions (1968), New York: Academic Press, New York · Zbl 0159.18301 |
[7] | Gröchenig, K., Foundations of Time-Frequency Analysis (2001), Boston: Birkhäuser, Boston · Zbl 0966.42020 · doi:10.1007/978-1-4612-0003-1 |
[8] | Hitrik, M.; Pravda-Starov, K., Spectra and semigroup smoothing for non-elliptic quadratic operators, Math. Ann., 344, 801-846 (2009) · Zbl 1171.47038 · doi:10.1007/s00208-008-0328-y |
[9] | Hörmander, L., The Analysis of Linear Partial Differential Operators (1990), Berlin: Springer, Berlin · Zbl 0687.35002 |
[10] | Hörmander, L.; Cattabriga, L.; Rodino, L., Quadratic hyperbolic operators, Microlocal Analysis and Applications, 118-160 (1991), Berlin: Springer, Berlin · Zbl 0761.35004 · doi:10.1007/BFb0085123 |
[11] | Hörmander, L., Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z., 219, 413-449 (1995) · Zbl 0829.35150 · doi:10.1007/BF02572374 |
[12] | Krantz, SG; Parks, HR, The Implicit Function Theorem (2003), Boston: Birkhäuser, Boston · Zbl 1269.58003 · doi:10.1007/978-1-4612-0059-8 |
[13] | Nakamura, S., Propagation of the homogeneous wave front set for Schrödinger equations, Duke Math. J., 126, 2, 349-367 (2005) · Zbl 1130.35023 · doi:10.1215/S0012-7094-04-12625-9 |
[14] | Nicola, F.; Rodino, L., Global Pseudo-Differential Calculus on Euclidean Spaces (2010), Basel: Birkhäuser, Basel · Zbl 1257.47002 · doi:10.1007/978-3-7643-8512-5 |
[15] | Parenti, C.; Rodino, L., Parametrices for a class of pseudo differential operators I, Ann. Mat. Pura Appl., 125, 4, 221-254 (1980) · Zbl 0406.35065 · doi:10.1007/BF01789413 |
[16] | Petersson, A.: Fourier characterizations and non-triviality of Gelfand-Shilov spaces, with applications to Toeplitz operators. arXiv:2202.00938 [math.FA] (2022) |
[17] | Pravda-Starov, K.; Rodino, L.; Wahlberg, P., Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians, Math. Nachr., 291, 1, 128-159 (2018) · Zbl 1384.35107 · doi:10.1002/mana.201600410 |
[18] | Reed, M.; Simon, B., Methods of Modern Mathematical Physics (1980), New York: Academic Press, New York · Zbl 0459.46001 |
[19] | Rodino, L., Wahlberg, P.: Anisotropic global microlocal analysis for tempered distributions. Monatsh. Math. (in press) (2022) |
[20] | Rodino, L., Wahlberg, P.: Microlocal analysis of Gelfand-Shilov spaces. arXiv:2202.05543 [math.AP] (2022) |
[21] | Rodino, L., Linear Partial Differential Operators in Gevrey Spaces (1993), Singapore: World Scientific, Singapore · Zbl 0869.35005 · doi:10.1142/1550 |
[22] | Rodino, L.; Wahlberg, P., The Gabor wave front set, Monaths. Math., 173, 4, 625-655 (2014) · Zbl 1366.42030 · doi:10.1007/s00605-013-0592-0 |
[23] | Schaefer, HH; Wolff, MP, Topological Vector Spaces (1999), New York: Springer, New York · Zbl 0983.46002 · doi:10.1007/978-1-4612-1468-7 |
[24] | Schulz, R.; Wahlberg, P., Equality of the homogeneous and the Gabor wave front set, Commun. PDE, 42, 5, 703-730 (2017) · Zbl 1373.35345 · doi:10.1080/03605302.2017.1300173 |
[25] | Toft, J., The Bargmann transform on modulation and Gelfand-Shilov spaces, with applications to Toeplitz and pseudo-differential operators, J. Pseudo-Differ. Oper. Appl., 3, 2, 145-227 (2012) · Zbl 1257.42033 · doi:10.1007/s11868-011-0044-3 |
[26] | Wahlberg, P., Propagation of polynomial phase space singularities for Schrödinger equations with quadratic Hamiltonians, Math. Scand., 122, 1, 107-140 (2018) · Zbl 1417.35155 · doi:10.7146/math.scand.a-97187 |
[27] | Zhu, H.: Propagation of singularities for gravity-capillary water waves. arXiv:1810.09339 [math.AP] (2020) |
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