Two weighted inequalities for fractional integrals on commutative hypergroups

In the present paper we prove a two weighted inequality for the fractional integrals on commutative hypergroups with quasi-metric and doubling Haar measure upper Ahlfors N-regular on an identity. This result is an generalization of the Heinig theorem obtained in [22], [6], [7] for the classical Riesz potential, for for the fractional integrals (B fractional integrals) associated with the Laplace-Bessel differential operator and for the fractional integrals on Laguerre hypergroups respectively. Also the SteinWeiss inequality for the fractional integrals on commutative hypergroups is proved as an application of the main result.