Riesz potentials derived by one-mode interacting Fock space approach

Nobuhiro Asai¹

¹ Department of Mathematics Education Aichi University of Education Kariya, 448-8542, Japan

Colloquium Mathematicum, Tome 109 (2007) no. 1, pp. 101-106.

Résumé

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 $\gamma_{\alpha,c_{1,2}}$.

Détail

DOI : 10.4064/cm109-1-8

Keywords: main short paper study riesz potentials one mode interacting fock spaces equipped deformed annihilation creation neutral operators constants mathbb geq equations first emphasize importance these constants summarize previous results hilbert space analytic functions respect probability measure mathbb consider riesz kernels order alpha alpha mathbb which derived bessel kernels order alpha gamma alpha mathbb moreover prove riesz potentials continuous linear operators hilbert space analytic functions respect gamma alpha