Riesz potentials derived by one-mode interacting Fock space approach
Volume 109 / 2007
Abstract
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 $\gamma_{\alpha,c_{1,2}}$.
