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  • 1.
    Cappiello, Marco
    et al.
    University of Turin, Italy.
    Schulz, René
    Universität Hannover, Germany.
    Wahlberg, Patrik
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Shubin type Fourier integral operators and evolution equations2019In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999XArticle in journal (Refereed)
    Abstract [en]

    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.

  • 2.
    Chen, Yuanyuan
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Strong ultra-regularity properties for positive elements in the twisted convolutions2017In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 8, no 4, p. 707-721Article in journal (Refereed)
    Abstract [en]

    We show that positive elements with respect to the twisted convolutions, belonging to some ultra-test function space of certain order at origin, belong to the ultra-test function space of the same order everywhere. We apply the result to positive semi-definite Weyl operators.

  • 3.
    Chen, Yuanyuan
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Boundedness of Gevrey and Gelfand-Shilov kernels of positive semi-definite operators2015In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 6, no 2, p. 153-185Article in journal (Refereed)
    Abstract [en]

    We show that the strongest Gevrey irregularity of kernels to positive semi-definite operators appear at the diagonals. We also prove that positive elements with respect to the twisted convolution, belonging to a Gevrey class of certain order at the origin, belong to the Gelfand-Shilov space of the same order. In the end we apply these results to positive semi-definite pseudo-differential operators.

  • 4.
    Coriasco, Sandro
    et al.
    Univ Turin, Italy.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Asymptotic expansions for Hörmander symbol classes in the calculus of pseudo-differential operators2014In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 5, no 1, p. 27-41Article in journal (Refereed)
    Abstract [en]

    We establish formulas for asymptotic expansions for S(m,g), the Hörmander class parameterized by the metric g and weight function m, defined on the phase space. By choosing m and g in appropriate ways, we cover some classical results on expansions for the standard symbol classes, and by choosing m and g in other ways we obtain asymptotic expansions for (generalized) SG classes.

  • 5.
    Fernandez, Carmen
    et al.
    Univ Valencia, Spain.
    Galbis, Antonio
    Univ Valencia, Spain.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Characterizations of GRS-weights, and consequences in time-frequency analysis2015In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 6, no 3, p. 383-390Article in journal (Refereed)
    Abstract [en]

    Let v be a submultiplicative weight. Then we prove that v satisfies Gel'fand-Raikov-Shilov-condition, if and only if is bounded for every positive . We use this equivalence to establish identification properties between weighted Lebesgue spaces, and between certain modulation spaces and Gelfand-Shilov spaces.

  • 6.
    Johansson, Karoline
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Association between temperate distributions and analytical functions in the context of wave-front sets2011In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 2, no 1, p. 65-89Article in journal (Refereed)
    Abstract [en]

    Let B be a translation invariant Banach function space (BF-space). In this paper we prove that every temperate distribution f can be associated with a function F analytic in the convex tube Ω = {z in Cd; | Im z| < 1 } such that the wave-front set of f of Fourier BF-space types in intersection with Rd ×Sd-1 consists of the points (x, ξ) such that F does not belong to the Fourier BF-space at xi ξ.

  • 7.
    Johansson, Karoline
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Generalized free time-dependent Schrödinger equation with initial data in Fourier Lebesgue spaces2011In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 2, no 4, p. 543-556Article in journal (Refereed)
    Abstract [en]

    Consider the solution of the free time-dependent Schrödinger equation with initial data f. It is shown by Sjögren and Sjölin that there exists f in the Sobolev spaceHs (Rd ), s = d/2 such that tangential convergence can not be widened to convergence regions. The author obtained in a previous paper the corresponding results for a generalized version of the Schrödinger equation, where −Δx is replaced by an operator ϕ(D), with special conditions on ϕ. In this paper we show that similar results may be obtained for initial data in usual and mixed Fourier Lebesgue spaces. We also relax the conditions on ϕ.

  • 8.
    Schulz, René
    et al.
    Leibniz Universität Hannover, Germany.
    Wahlberg, Patrik
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Microlocal properties of Shubin pseudodifferential and localization operators2016In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 7, no 1, p. 91-111Article in journal (Refereed)
    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.

  • 9.
    Signahl, Mikael
    et al.
    Univ Agder, Norway.
    Toft, Joachim
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    Mapping properties for the Bargmann transform on modulation spaces2012In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 3, no 1, p. 1-30Article in journal (Refereed)
    Abstract [en]

    We investigate the mapping properties for the Bargmann transform and prove that this transform is isometricand bijective from modulation spaces to convenient Banach spaces of analytic functions.

  • 10.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Continuity of Gevrey-Hörmander pseudo-differential operators on modulation spaces2019In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 10, no 2, p. 337-358Article in journal (Refereed)
    Abstract [en]

    Let s ≥ 1, ω,ω ∈ P^0_{E,s} , a ∈ 􏰒\Gamma _s^(ω_0), and let B be a suitable invariant quasi-Banach function space. Then we prove that the pseudo-differential operator Op(a) is continuous from M(ω_0·ω, B) to M(ω, B).

  • 11.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Images of function and distribution spaces under the Bargmann transform2017In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 8, no 1, p. 83-139Article in journal (Refereed)
    Abstract [en]

    Weconsiderabroadfamilyoftestfunctionspacesandtheirdual(distribu- tion) spaces. The family includes Gelfand–Shilov spaces, and a family of test function spaces introduced by Pilipovic ́. We deduce different characterizations of such spaces, especially under the Bargmann transform and the Short-time Fourier transform. The family also include a test function space, whose dual space is mapped by the Bargmann transform bijectively to the set of entire functions. 

  • 12.
    Toft, Joachim
    Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
    The Bargmann transform on modulation and Gelfand-Shilov spaces, with applications to Toeplitz and pseudo-differential operators2012In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 3, no 2, p. 145-227Article in journal (Refereed)
    Abstract [en]

    We investigate mapping properties for the Bargmann transform on an extended family ofmodulation spaces whose weights and their reciprocals are allowed to grow faster than exponentials. We provethat this transform is isometric and bijective from modulation spacesto convenient Lebesgue spaces of analytic functions. We use this to prove that suchmodulation spaces fulfill most of the continuity properties which are valid for modulationspaces with moderate weights. Finally we use the results to establish continuity propertiesof Toeplitz and pseudo-differential operators on these modulation spaces, and onGelfand-Shilov spaces.

  • 13.
    Wahlberg, Patrik
    Department of Mathematics, University of Turin, Via Carlo Alberto 10, 10123, Turin, Italy.
    Representations of almost periodic pseudodifferential operators and applications in spectral theory2012In: Journal of Pseudo-Differential Operators and Applications, ISSN 1662-9981, E-ISSN 1662-999X, Vol. 3, no 1, p. 81-119Article in journal (Refereed)
    Abstract [en]

    The paper concerns algebras of almost periodic pseudodifferential operators on Rd with symbols in Hörmander classes. We study three representations of such algebras, one of which was introduced by Coburn, Moyer and Singer and the other two inspired by results in probability theory by Gladyshev. Two of the representations are shown to be unitarily equivalent for nonpositive order. We apply the results to spectral theory for almost periodic pseudodifferential operators acting on L 2 and on the Besicovitch Hilbert space of almost periodic functions.

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