Two weighted inequalities for fractional integrals on Laguerre hypergroup

摘要

Let K = [0,∞) × R be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group, | · | its homogeneous norm and Q its homogeneous dimension. In this paperweprove the two weighted inequality for fractional integrals Iβ on K. The obtained result is an analog of the Heinig result [Heinig HP. Weighted norm inequalities for classes of operators. Indiana Univ Math J. 1984;33(4):573-582] for fractional integrals on Laguerre hypergroup. Furthermore, the Stein-Weiss inequality for Iβ is proved as an application of this result.