What is a Sobolev space for the Laguerre function systems?
Volume 192 / 2009
Studia Mathematica 192 (2009), 147-172
MSC: Primary 42B35; Secondary 42C05.
DOI: 10.4064/sm192-2-4
Abstract
We discuss the concept of Sobolev space associated to the Laguerre operator $ L_\alpha = - y\,\frac{d^2}{dy^2} - \frac{d}{dy} + \frac{y}{4} + \frac{\alpha^2}{4y},\ y\in (0,\infty).$ We show that the natural definition does not agree with the concept of potential space defined via the potentials $ (L_\alpha)^{-s}.$ An appropriate Laguerre–Sobolev space is defined in order to achieve that coincidence. An application is given to the almost everywhere convergence of solutions of the Schrödinger equation. Other Laguerre operators are also considered.
