What is a Sobolev space for the Laguerre function systems?

Abstract

We discuss the concept of Sobolev space associated to the Laguerre operator $ L_\al = - y\,\frac{d^2}{dy^2} - \frac{d}{dy} + \frac{y}{4} + \frac{\al^2}{4y},\quad y\in (0,\infty).$ We show that the natural definition does not fit with the concept of potential space, defined via the potentials $ (L_\al)^{-s}.$ An appropriate Laguerre-Sobolev spaces are defined in order to have the mentioned coincidence. An application is given to the almost everywhere convergence of solutions of the Schr\odinger equation. Other Laguerre operators are also considered. Published: Studia Math. 192 (2009), no. 2, 147--172.