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  • Research Article
  • Open Access

Inequalities for differentiable reproducing kernels and an application to positive integral operators

Journal of Inequalities and Applications20062006:53743

  • Received: 18 October 2005
  • Accepted: 13 November 2005
  • Published:


Let be an interval and let be a reproducing kernel on . We show that if is in the appropriate differentiability class, it satisfies a 2-parameter family of inequalities of which the diagonal dominance inequality for reproducing kernels is the 0th order case. We provide an application to integral operators: if is a positive definite kernel on (possibly unbounded) with differentiability class and satisfies an extra integrability condition, we show that eigenfunctions are and provide a bound for its Sobolev norm. This bound is shown to be optimal.


  • Integral Operator
  • Integrability Condition
  • Sobolev Norm
  • Order Case
  • Diagonal Dominance


Authors’ Affiliations

Departamento de Matemática, Instituto Superior Técnico, Lisbon, 1049-001, Portugal
Departamento de Engenharia Mecânica, ISEL, Lisbon, 1949-014, Portugal


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© J. Buescu and A. C. Paix˜ao. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.