| 1. |
- Boggiatto, Paolo, et al.
(author)
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The wave front set of the Wigner distribution and instantaneous frequency
- 2012
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In: Journal of Fourier Analysis and Applications. - Boston : Birkhäuser Verlag. - 1069-5869 .- 1531-5851. ; 18:2, s. 410-438
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Journal article (peer-reviewed)abstract
- We prove a formula expressing the gradient of the phase function of a function f : R-d bar right arrow C as a normalized first frequency momentof the Wigner distribution for fixed time. The formula holds when f is the Fourier transform of a distribution of compact support, or when f belongs to a Sobolev space Hd/2+1+epsilon(R-d) where epsilon > 0. The restriction of the Wigner distribution to fixed time is well defined provided a certain condition on its wave front set is satisfied. Therefore we first need to study the wave front set of the Wigner distribution of a tempered distribution.
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| 2. |
- Cappiello, Marco, et al.
(author)
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Conormal distributions in the Shubin calculus of pseudodifferential operators
- 2018
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In: Journal of Mathematical Physics. - : American Institute of Physics (AIP). - 0022-2488 .- 1089-7658. ; 59:2
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Journal article (peer-reviewed)abstract
- We characterize the Schwartz kernels of pseudodifferential operators of Shubin type by means of a Fourier-Bros-Iagolnitzer transform. Based on this, we introduce as a generalization a new class of tempered distributions called Shubin conormal distributions. We study their transformation behavior, normal forms, and microlocal properties.
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| 3. |
- Cappiello, Marco, et al.
(author)
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Shubin type Fourier integral operators and evolution equations
- 2020
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In: Journal of Pseudo-Differential Operators and Applications. - : Springer. - 1662-9981 .- 1662-999X. ; 11:1, s. 119-139
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Journal article (peer-reviewed)abstract
- We study the Cauchy problem for an evolution equation of Schrödinger type. The Hamiltonian is the Weyl quantization of a real homogeneous quadratic form with a pseudodifferential perturbation of negative order from Shubin’s class. We prove that the propagator is a Fourier integral operator of Shubin type of order zero. Using results for such operators and corresponding Lagrangian distributions, we study the propagator and the solution, and derive phase space estimates for them.
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| 4. |
- Carypis, Evanthia, et al.
(author)
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Propagation of exponential phase space singularities for Schrödinger equations with quadratic Hamiltonians
- 2017
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In: Journal of Fourier Analysis and Applications. - : Springer. - 1069-5869 .- 1531-5851. ; 23:3, s. 530-571
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Journal article (peer-reviewed)abstract
- We study propagation of phase space singularities for the initial value Cauchy problem for a class of Schrödinger equations. The Hamiltonian is the Weyl quantization of a quadratic form whose real part is non-negative. The equations are studied in the framework of projective Gelfand–Shilov spaces and their distribution duals. The corresponding notion of singularities is called the Gelfand–Shilov wave front set and means the lack of exponential decay in open cones in phase space. Our main result shows that the propagation is determined by the singular space of the quadratic form, just as in the framework of the Schwartz space, where the notion of singularity is the Gabor wave front set.
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| 5. |
- Chen, Yuanyuan, et al.
(author)
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The Weyl product on quasi-Banach modulation spaces
- 2019
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In: Bulletin of Mathematical Sciences. - : World Scientific. - 1664-3607 .- 1664-3615. ; 9:2, s. 1-30
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Journal article (peer-reviewed)abstract
- We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Banach modulation spaces.
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| 6. |
- Cordero, Elena, et al.
(author)
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Schrodinger-type propagators, pseudodifferential operators and modulation spaces
- 2013
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In: Journal of the London Mathematical Society. - London : London Mathematical Society. - 0024-6107 .- 1469-7750. ; 88, s. 375-395
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Journal article (peer-reviewed)abstract
- We prove continuity results for Fourier integral operators with symbols in modulation spaces, acting between modulation spaces. The phase functions belong to a class of non-degenerate generalized quadratic forms that includes Schrödinger propagators and pseudodifferential operators. As a byproduct, we obtain a characterization of all exponents p, q, r1, r2, t1, t2∈[1, ∞] of modulation spaces such that a symbol in Mp, q(ℝ2d) gives a pseudodifferential operator that is continuous from Mr1,r2(ℝd) into Mt1,t2(ℝd).
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| 7. |
- Cordero, Elena, et al.
(author)
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Sharp results for the Weyl product on modulation spaces
- 2014
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In: Journal of Functional Analysis. - : Elsevier. - 0022-1236 .- 1096-0783. ; 267:8, s. 3016-3057
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Journal article (peer-reviewed)abstract
- We give sufficient and necessary conditions on the Lebesgue exponentsfor the Weyl product to be bounded on modulation spaces. The sufficient conditions are obtained as the restriction to N=2 of aresult valid for the N-fold Weyl product. As a byproduct, we obtain sharpconditions for the twisted convolution to be bounded on Wieneramalgam spaces.
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| 8. |
- Holst, Anders, et al.
(author)
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Weyl product algebras and classical modulation spaces
- 2010
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In: <em>Linear and non-linear theory of generalized functions and its applications</em>. - Warsaw : Polish Acad. Sci. Inst. Math.. - 9788386806072 ; , s. 153-158
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Conference paper (peer-reviewed)abstract
- We discuss continuity properties of the Weyl product when acting on classical modulation spaces. In particular, we prove that M p,q is an algebra under the Weyl product when p∈[1,∞] and 1≤q≤min(p,p ′ ) .
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| 9. |
- Oliaro, Alessandro, et al.
(author)
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Almost periodic pseudodifferential operators and Gevrey classes
- 2012
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In: Annali di Matematica Pura ed Applicata. - : Springer. - 0373-3114 .- 1618-1891. ; 191:4, s. 725-760
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Journal article (peer-reviewed)abstract
- We study almost periodic pseudodifferential operators acting on almost periodic functions G s ap (R d ) of Gevrey regularity index s ≥ 1. We prove that almost periodic operators with symbols of Hörmander type S m ρ,δ satisfying an s-Gevrey condition are continuous on G s ap (R d ) provided 0 < ρ ≤ 1, δ = 0 and s ρ ≥ 1. A calculus is developed for symbols and operators using a notion of regularizing operator adapted to almost periodic Gevrey functions and its duality. We apply the results to show a regularity result in this context for a class of hypoelliptic operators.
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| 10. |
- Pravda-Starov, Karel, et al.
(author)
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Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians
- 2018
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In: Mathematische Nachrichten. - : Wiley-Blackwell. - 0025-584X .- 1522-2616. ; 291:1, s. 128-159
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Journal article (peer-reviewed)abstract
- We study propagation of the Gabor wave front set for a Schrödinger equation wit ha Hamiltonian that is the Weyl quantization of a quadratic form with nonnegativereal part. We point out that t he singular space associated with the quadratic formplays a crucial role for the understanding of this propagation. We show that the Gaborsingularities of the solution to the equation for positive times are always contained inthe singular space, and that t hey propagate in this set along the flow of the Hamiltonvector field associated with the imaginary part of the quadratic form. As an applicationwe obtain for the heat equation a sufficient condition on the Gabor wave front set of theinitial datum tempered distribution that implies regularization to Schwartz regularityfor positive times.
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