Eigenvalues of Hille–Tamarkin operators and geometry of Banach function spaces

Volume 207 / 2011

Thomas Kühn, Mieczysław Mastyło Studia Mathematica 207 (2011), 275-296 MSC: Primary 47B06, 47G10; Secondary 47B10, 46E30. DOI: 10.4064/sm207-3-4


We investigate how the asymptotic eigenvalue behaviour of Hille–Tamarkin operators in Banach function spaces depends on the geometry of the spaces involved. It turns out that the relevant properties are cotype $p$ and $p$-concavity. We prove some eigenvalue estimates for Hille–Tamarkin operators in general Banach function spaces which extend the classical results in Lebesgue spaces. We specialize our results to Lorentz, Orlicz and Zygmund spaces and give applications to Fourier analysis. We are also able to show the optimality of our eigenvalue estimates in the Lorentz spaces $L_{2,q}$ with $1\le q<2$ and in Zygmund spaces $L_p(\log L)_a$ with $2\le p<\infty$ and $a>0$.


  • Thomas KühnFakultät für Mathematik und Informatik
    Mathematisches Institut
    Universität Leipzig
    Johannisgasse 26
    D-04103 Leipzig, Germany
  • Mieczysław MastyłoFaculty of Mathematics and Computer Science
    Adam Mickiewicz University
    Institute of Mathematics
    Polish Academy of Sciences (Poznań branch)
    Umultowska 87
    61-614 Poznań, Poland

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