Suitable domains to define fractional integrals of Weyl via fractional powers of operators

Celso Martínez¹ ; Antonia Redondo² ; Miguel Sanz¹

¹ Departament de Matemàtica Aplicada Universitat de València 46100 Burjassot, València, Spain
² Departamento de Matemáticas I.E.S. Bachiller Sabuco 02002 Albacete, Spain

Studia Mathematica, Tome 202 (2011) no. 2, pp. 145-164

Résumé

We present a new method to study the classical fractional integrals of Weyl. This new approach basically consists in considering these operators in the largest space where they make sense. In particular, we construct a theory of fractional integrals of Weyl by studying these operators in an appropriate Fréchet space. This is a function space which contains the $L^{p}( \mathbb{R})$-spaces, and it appears in a natural way if we wish to identify these fractional operators with fractional powers of a suitable non-negative operator. This identification allows us to give a unified view of the theory and provides some elegant proofs of some well-known results on the fractional integrals of Weyl.

Détail

DOI : 10.4064/sm202-2-3

Keywords: present method study classical fractional integrals weyl approach basically consists considering these operators largest space where make sense particular construct theory fractional integrals weyl studying these operators appropriate chet space function space which contains mathbb spaces appears natural wish identify these fractional operators fractional powers suitable non negative operator identification allows unified view theory provides elegant proofs well known results fractional integrals weyl