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  • 1.
    Chen, Yuanyuan
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Continuity and compositions of operators with kernels in ultra-test function and ultra-distribution spaces2016Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    In this thesis we consider continuity and positivity properties of pseudo-differential operators in Gelfand-Shilov and Pilipović spaces, and their distribution spaces. We also investigate composition property of pseudo-differential operators with symbols in quasi-Banach modulation spaces.

    We prove that positive elements with respect to the twisted convolutions, possesing Gevrey regularity of certain order at origin, belong to the Gelfand-Shilov space of the same order. We apply this result to positive semi-definite pseudo-differential operators, as well as show that the strongest Gevrey irregularity of kernels to positive semi-definite operators appear at the diagonals.

    We also prove that any linear operator with kernel in a Pilipović or Gelfand-Shilov space can be factorized by two operators in the same class. We give links on numerical approximations for such compositions and apply these composition rules to deduce estimates of singular values and establish Schatten-von Neumann properties for such operators.  

    Furthermore, we derive sufficient and necessary conditions for continuity of the Weyl product with symbols in quasi-Banach modulation spaces.

  • 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.
    Signahl, Mikael
    University of Agder, Norway.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Factorizations and singular value estimates of operators with Gelfand-Shilov and Pilipovic' kernels2018In: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 24, no 3, p. 666-698Article in journal (Refereed)
    Abstract [en]

    We prove that any linear operator with kernel in a Pilipovi{\'c} or Gelfand-Shilov spacecan be factorized by two operators in the same class. We also give links onnumerical approximations for such compositions. We apply these composition rulesto deduce estimates of singular values and establish Schatten-von Neumann propertiesfor such operators.

  • 4.
    Chen, Yuanyuan
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Signahl, Mikael
    University of Agder, Norway.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Hilbert space embeddings for Gelfand–Shilov and Pilipović spaces2017In: Generalized Functions and Fourier Analysis: Dedicated to Stevan Pilipović on the Occasion of his 65th Birthday / [ed] Michael Oberguggenberger, Joachim Toft, Jasson Vindas, Patrik Wahlberg, Springer, 2017, p. 31-44Chapter in book (Refereed)
    Abstract [en]

    We consider quasi-Banach spaces that lie between a Gelfand–Shilov space, or more generally, Pilipovi´c space, H, and its dual, H′. We prove that for such quasi-Banach space B, there are convenient Hilbert spaces, Hk, k=1,2ss, with normalized Hermite functions as orthonormal bases and such that B lies between H1 and H1, and the latter spaces lie between H and H′.

  • 5.
    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.

  • 6.
    Chen, Yuanyuan
    et al.
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Toft, Joachim
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    Wahlberg, Patrik
    Linnaeus University, Faculty of Technology, Department of Mathematics.
    The Weyl product on quasi-Banach modulation spaces2019In: Bulletin of Mathematical Sciences, ISSN 1664-3607, E-ISSN 1664-3615, Vol. 9, no 2, p. 1-30, article id 1950018Article in journal (Refereed)
    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.

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