Reproducing properties of differentiable Mercer-like kernels

Let X be an open subset of R-d and nu the restriction of the usual Lebesgue measure of R-d to X. In this paper, we investigate properties of the range of positive integral operators on L-2(X, nu), in connection with the reproducing kernel Hilbert space of the generating kernel. Assuming differentiability assumptions on the kernel, we deduce smoothness properties for the functions in the range of the operator and also properties of the so-called inclusion map. The results are deduced when the assumptions are defined via both, weak and partial derivatives. Further, assuming the generating kernel has a Mercer-like expansion based on sufficiently smooth functions, we deduce results on the term-by-term differentiability of the series and reproducing properties for the derivatives of the functions in the reproducing kernel Hilbert space.