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