Inequalities for Differentiable Reproducing Kernels and an Application to Positive Integral Operators

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 n(I) 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.