A choice of Sobolev spaces associated with ultraspherical expansions

BETANCOR, JORGE J.; FARIÑA, JUAN C.; RODRÍGUEZ-MESA, LOURDES; TESTONI, RICARDO; TORREA, JOSÉ L.

PUBLICACIONS MATEMATIQUES

UNIV AUTONOMA BARCELONA

Lugar: Barcelona; Año: 2010 vol. 54 p. 221 - 242

0214-1493

We discuss two possible definitions for Sobolev spaces associatedwith ultraspherical expansions. These definitions depend on thenotion of higher order derivative. We show that in order to havean isomorphism between Sobolev and potential spaces, the higherorder derivatives to be considered are not the iteration of thefirst order derivatives. Some discussions about higher order Riesztransforms are involved. Also we prove that the maximal operatorfor the Poisson integral in the ultraspherical setting is boundedon the Sobolev spaces.