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Wahlberg, Patrik
Publications (10 of 32) Show all publications
Cappiello, M., Schulz, R. & Wahlberg, P. (2018). Conormal distributions in the Shubin calculus of pseudodifferential operators. Journal of Mathematical Physics, 59(2), Article ID 021502.
Open this publication in new window or tab >>Conormal distributions in the Shubin calculus of pseudodifferential operators
2018 (English)In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 59, no 2, article id 021502Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
American Institute of Physics (AIP), 2018
National Category
Mathematical Analysis
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-70554 (URN)10.1063/1.5022778 (DOI)000426583800002 ()
Available from: 2018-02-07 Created: 2018-02-07 Last updated: 2018-03-22Bibliographically approved
Pravda-Starov, K., Rodino, L. & Wahlberg, P. (2018). Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians. Mathematische Nachrichten, 291(1), 128-159
Open this publication in new window or tab >>Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians
2018 (English)In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 291, no 1, p. 128-159Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Wiley-Blackwell, 2018
National Category
Mathematical Analysis
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-61660 (URN)10.1002/mana.201600410 (DOI)000419960200010 ()
Available from: 2017-03-23 Created: 2017-03-23 Last updated: 2018-02-08Bibliographically approved
Wahlberg, P. (2018). Propagation of polynomial phase space singularities for Schrödinger equations with quadratic Hamiltonians. Mathematica Scandinavica, 122(1), 107-140
Open this publication in new window or tab >>Propagation of polynomial phase space singularities for Schrödinger equations with quadratic Hamiltonians
2018 (English)In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 122, no 1, p. 107-140Article in journal (Refereed) Published
Abstract [en]

We study propagation of phase space singularities for a Schrödinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with non-negative real part. Phase space singularities are measured by the lack of polynomial decay of given order in open cones in the phase space, which gives a parametrized refinement of the Gabor wave front set. The main result confirms the fundamental role of the singular space associated to the quadratic form for the propagation of phase space singularities. The singularities are contained in the singular space, and propagate in the intersection of the singular space and the initial datum singularities along the flow of the Hamilton vector field associated to the imaginary part of the quadratic form.

National Category
Mathematical Analysis
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-51647 (URN)10.7146/math.scand.a-97187 (DOI)000448483600007 ()
Available from: 2016-03-30 Created: 2016-03-30 Last updated: 2018-11-08Bibliographically approved
Chen, Y., Toft, J. & Wahlberg, P. (2018). The Weyl product on quasi-Banach modulation spaces. Bulletin of Mathematical Sciences
Open this publication in new window or tab >>The Weyl product on quasi-Banach modulation spaces
2018 (English)In: Bulletin of Mathematical Sciences, ISSN 1664-3607, E-ISSN 1664-3615Article in journal (Refereed) Epub ahead of print
Abstract [en]

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.

National Category
Mathematical Analysis
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-70220 (URN)10.1007/s13373-018-0116-2 (DOI)
Available from: 2018-01-29 Created: 2018-01-29 Last updated: 2018-12-11
Carypis, E. & Wahlberg, P. (2017). Propagation of exponential phase space singularities for Schrödinger equations with quadratic Hamiltonians. Journal of Fourier Analysis and Applications, 23(3), 530-571
Open this publication in new window or tab >>Propagation of exponential phase space singularities for Schrödinger equations with quadratic Hamiltonians
2017 (English)In: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 23, no 3, p. 530-571Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Springer, 2017
National Category
Mathematical Analysis
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-51970 (URN)10.1007/s00041-016-9478-6 (DOI)000401411900002 ()
Available from: 2016-04-08 Created: 2016-04-08 Last updated: 2017-07-18Bibliographically approved
Schulz, R. & Wahlberg, P. (2017). The Equality of the homogeneous and the Gabor wave front set. Communications in Partial Differential Equations, 42(5), 703-730
Open this publication in new window or tab >>The Equality of the homogeneous and the Gabor wave front set
2017 (English)In: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 42, no 5, p. 703-730Article in journal (Refereed) Published
Abstract [en]

We prove that Hörmander’s global wave front set and Nakamura’s homogeneous wave front set of a tempered distribution coincide. In addition we construct a tempered distribution with a given wave front set, and we develop a pseudodifferential calculus adapted to Nakamura’s homogeneous wave front set.

Place, publisher, year, edition, pages
Taylor & Francis, 2017
National Category
Mathematical Analysis
Research subject
Mathematics, Mathematics; Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-61659 (URN)10.1080/03605302.2017.1300173 (DOI)000400513700002 ()
Available from: 2017-03-23 Created: 2017-03-23 Last updated: 2017-05-29Bibliographically approved
Schulz, R. & Wahlberg, P. (2016). Microlocal properties of Shubin pseudodifferential and localization operators. Journal of Pseudo-Differential Operators and Applications, 7(1), 91-111
Open this publication in new window or tab >>Microlocal properties of Shubin pseudodifferential and localization operators
2016 (English)In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 7, no 1, p. 91-111Article in journal (Refereed) Published
Abstract [en]

We investigate global microlocal properties of localization operators and Shubin pseudodifferential operators. The microlocal regularity is measured in terms of a scale of Shubin-type Sobolev spaces. In particular, we prove microlocality and microellipticity of these operators.

National Category
Mathematical Analysis
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-49573 (URN)10.1007/s11868-015-0143-7 (DOI)000376460600007 ()2-s2.0-84959346728 (Scopus ID)
Available from: 2016-02-04 Created: 2016-02-04 Last updated: 2017-11-30Bibliographically approved
Cordero, E., Toft, J. & Wahlberg, P. (2014). Sharp results for the Weyl product on modulation spaces. Journal of Functional Analysis, 267(8), 3016-3057
Open this publication in new window or tab >>Sharp results for the Weyl product on modulation spaces
2014 (English)In: Journal of Functional Analysis, ISSN 0022-1236, E-ISSN 1096-0783, Vol. 267, no 8, p. 3016-3057Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
Elsevier, 2014
Keywords
Weyl product; modulation spaces; twisted convolution; sharpness
National Category
Mathematical Analysis
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-36031 (URN)10.1016/j.jfa.2014.07.011 (DOI)000347745900015 ()2-s2.0-84908591702 (Scopus ID)
Projects
matematisk modellering
Available from: 2014-07-11 Created: 2014-07-11 Last updated: 2017-12-05Bibliographically approved
Rodino, L. & Wahlberg, P. (2014). The Gabor wave front set. Monatshefte für Mathematik (Print), 173(4), 625-655
Open this publication in new window or tab >>The Gabor wave front set
2014 (English)In: Monatshefte für Mathematik (Print), ISSN 0026-9255, E-ISSN 1436-5081, Vol. 173, no 4, p. 625-655Article in journal (Refereed) Published
Abstract [en]

We define the Gabor wave front set W F-G(u) of a tempered distribution u in terms of rapid decay of its Gabor coefficients in a conic subset of the phase space. We show the inclusion W F-G(a(w) (x, D)u) subset of W F-G(u), u is an element of l'(R-d), a is an element of S-0,0(0), where S-0,0(0) denotes the Hormander symbol class of order zero and parameter values zero. We compare our definition with other definitions in the literature, namely the classical and the global wave front sets of Hormander, and the l-wave front set of Coriasco and Maniccia. In particular, we prove that the Gabor wave front set and the global wave front set of Hormander coincide.

National Category
Mathematical Analysis
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-32273 (URN)10.1007/s00605-013-0592-0 (DOI)000333161400011 ()2-s2.0-84896393828 (Scopus ID)
Available from: 2014-02-12 Created: 2014-02-12 Last updated: 2017-12-06Bibliographically approved
Cordero, E., Tabacco, A. & Wahlberg, P. (2013). Schrodinger-type propagators, pseudodifferential operators and modulation spaces. Journal of the London Mathematical Society, 88, 375-395
Open this publication in new window or tab >>Schrodinger-type propagators, pseudodifferential operators and modulation spaces
2013 (English)In: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. 88, p. 375-395Article in journal (Refereed) Published
Abstract [en]

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).

Place, publisher, year, edition, pages
London: London Mathematical Society, 2013
National Category
Mathematical Analysis
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-27694 (URN)10.1112/jlms/jdt020 (DOI)
Available from: 2013-07-29 Created: 2013-07-29 Last updated: 2017-12-06Bibliographically approved
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