  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, which couple the regularity of the driving
noise with the properties of the differential operator, have important
implications for stochastic partial differential equations on bounded domains
as well as their numerical approximations. 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-homogeneous and homogeneous kernels are considered. In the latter case,
results in a general Schatten class norm are also provided. Important examples
of homogeneous kernels covered by the results of the paper include the class of
Mat\'ern kernels.
