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

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.