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  • 1.
    Eskandari, Setareh
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran.
    Abkar, Ali
    Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran.
    Åhag, Per
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Perälä, Antti
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Sub-Hilbert relation for Fock-Sobolev type spaces2022In: New York Journal of Mathematics, E-ISSN 1076-9803, Vol. 28, p. 958-969Article in journal (Refereed)
    Abstract [en]

    In this paper, two specific sub-Hilbert spaces are studied. They arise from the action of a Toeplitz operator on Fock–Sobolev type spaces, in-duced by a general Gaussian type weight. The argument is based on analysing the reproducing kernel of the corresponding sub-Hilbert space.

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  • 2. Liu, Congwen
    et al.
    Perälä, Antti
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Si, Jiajia
    Weighted integrability of polyharmonic functions in the higher-dimensional case2021In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206X, Vol. 14, no 7, p. 2047-2068Article in journal (Refereed)
    Abstract [en]

    This paper is concerned with the L-p integrability of N-harmonic functions with respect to the standard weights (1 - vertical bar x vertical bar(2))(alpha) on the unit ball B of R-n, n >= 2. More precisely, our goal is to determine the real (negative) parameters alpha for which (1 - vertical bar x vertical bar(2))(alpha/p) u(x) is an element of L-p(B) implies that u equivalent to 0 whenever u is a solution of the N-Laplace equation on B. This question is motivated by the uniqueness considerations of the Dirichlet problem for the N-Laplacian Delta(N).

    Our study is inspired by a recent work of Borichev and Hedenmalm (Adv. Math. 264 (2014), 464-505), where a complete answer to the above question in the case n D 2 is given for the full scale 0 < p < infinity. When n >= 3, we obtain an analogous characterization for n-2/n-1 <= p < infinity and remark that the remaining case can be genuinely more difficult. Also, we extend the remarkable cellular decomposition theorem of Borichev and Hedenmalm to all dimensions.

  • 3.
    Pang, Changbao
    et al.
    School of Mathematics and Computer Science, Shanxi Normal University, Linfen, China.
    Perälä, Antti
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Wang, Maofa
    School of Mathematics and Statistics, Wuhan University, Wuhan, China.
    Embedding Theorems and Area Operators on Bergman Spaces with Doubling Measure2021In: Complex Analysis and Operator Theory, ISSN 1661-8254, E-ISSN 1661-8262, Vol. 15, no 3, article id 42Article in journal (Refereed)
    Abstract [en]

    We establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure dω(y) dx with the doubling property ω(0 , 2 t) ≤ Cω(0 , t). The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

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  • 4. Pang, Changbao
    et al.
    Perälä, Antti
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Wang, Maofa
    Guo, Xin
    Maximal estimate and integral operators in Bergman spaces with doubling measure2023In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 151, no 7, p. 2881-2894Article in journal (Refereed)
    Abstract [en]

    The boundedness of the maximal operator on the upper half-plane pi+ is established. Here pi+ is equipped with a positive Borel measure d omega(y)dx satisfying the doubling property omega ((0, 2t)) <= C omega ((0, t)). This result is connected to the Carleson embedding theorem, which we use to characterize the boundedness and compactness of the Volterra type integral operators on the Bergman spaces Ap omega(pi+).

  • 5.
    Perälä, Antti
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Virtanen, Jani
    Department of Mathematics and Statistics, University of Reading, Reading, UK; Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland.
    Essential positivity2023In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 151, no 11, p. 4807-4815Article in journal (Refereed)
    Abstract [en]

    We define essentially positive operators on Hilbert space as a class of self-adjoint operators whose essential spectra is contained in the non-negative real numbers and describe their basic properties. Using Toeplitz operators and the Berezin transform, we further illustrate the notion of essential positivity in the Hardy space and the Bergman space.

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