Hilbert--Schmidt regularity of symmetric integral operators on bounded domains with applications to SPDE approximations

Regularity estimates for an integral operator with a symmetric continuous kernel on a convex bounded domain are derived. The covariance of a mean-square continuous random field on the domain is an example of such an operator. The estimates are of the form of Hilbert--Schmidt norms of the integral operator and its square root, composed with fractional powers of an elliptic operator equipped with homogeneous boundary conditions of either Dirichlet or Neumann type. These types of estimates have important implications for stochastic partial differential equations on bounded domains as well as their numerical approximations, which couple the regularity of the driving noise with the properties of the differential operator. The main tools used to derive the estimates are properties of reproducing kernel Hilbert spaces of functions on bounded domains along with Hilbert--Schmidt embeddings of Sobolev spaces. Both non-homogenenous and homogeneous kernels are considered. Important examples of homogeneous kernels covered by the results of the paper include the class of Mat\'ern kernels.