What is a Sobolev space for the Laguerre function systems?

B. BONGIOANNI, J.L. TORREA

STUDIA MATHEMATICA

Año: 2008

0039-3223

We discuss the concept of Sobolev space associated to the Laguerre operator$ L_al = - y,rac{d^2}{dy^2} - rac{d}{dy} + rac{y}{4} + rac{al^2}{4y},quad yin (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.