Riesz potentials derived by one-mode interacting Fock space approach

Volume 109 / 2007

Nobuhiro Asai Colloquium Mathematicum 109 (2007), 101-106 MSC: 46N30, 33D45, 60J45. DOI: 10.4064/cm109-1-8


The main aim of this short paper is to study Riesz potentials on one-mode interacting Fock spaces equipped with deformed annihilation, creation, and neutral operators with constants $c_{0,0}, c_{1,1}\in {\mathbb R}$ and $c_{0,1}>0$, $c_{1,2}\geq 0$ as in equations (1.4)–(1.6). First, to emphasize the importance of these constants, we summarize our previous results on the Hilbert space of analytic $L^2$ functions with respect to a probability measure on $\mathbb C$. Then we consider the Riesz kernels of order $2\alpha$, $\alpha=c_{0,1}/c_{1,2},$ on $\mathbb C$ if $0< c_{0,1}< c_{1,2}$, which can be derived from the Bessel kernels of order $2\alpha$, $\gamma_{\alpha,c_{1,2}}$, on $\mathbb C$. Moreover, we prove that if $c_{1,2}/2< c_{0,1}< c_{1,2}$, then the Riesz potentials are continuous linear operators on the Hilbert space of analytic $L^2$ functions with respect to $\gamma_{\alpha,c_{1,2}}$.


  • Nobuhiro AsaiDepartment of Mathematics Education
    Aichi University of Education
    Kariya, 448-8542, Japan

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