Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 53743, 9 pages
doi:10.1155/JIA/2006/53743

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

Jorge Buescu1 and A. C. Paixão2

1Departamento de Matemática, Instituto Superior Técnico, Lisbon 1049-001, Portugal
2Departamento de Engenharia Mecânica, ISEL, Lisbon 1949-014, Portugal

Received 18 October 2005; Revised 7 November 2005; Accepted 13 November 2005

Copyright © 2006 Jorge Buescu and A. C. Paixão. 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.

Abstract

Let I⊆ℝ be an interval and let k:I2→ℂ be a reproducing kernel on I. We show that if k(x,y) 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 k is a positive definite kernel on I (possibly unbounded) with differentiability class 𝒮n(I2) and satisfies an extra integrability condition, we show that eigenfunctions are Cn(I) and provide a bound for its Sobolev Hn norm. This bound is shown to be optimal.