Hardy integral estimates for a class of integral transforms

Abstract

For a class of integral transforms T _φ over ℝ ⁿ , a sufficient condition is given which ensures that T _φ is bounded between weighted Lebesgue spaces. We also obtain a necessary and sufficient condition when the Hardy integral transform H of one of the weights is doubling. In such a case, the boundedness of T _φ and the boundedness of H are equivalent. The class of transforms considered here includes the Stieltjes transform, the Laplace transform, and the Lambert transform over ℝ ⁿ .

Keywords:

• weighted inequalities
• Hardy's inequality
• Hardy integral
• integral operator
• integral transform
• doubling condition

2000 Mathematics Subject classification :

• Primary 44A05
• Secondary 44A15
• 26D10

Acknowledgements

This research is supported in part by I-Shou University, Kaohsiung, ROC, under Grant ISU98-02-15.