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.

翻译：计算出一个具有对称连续内核的整合操作员在连接的域上的常态性估计值。一个平均平方连续随机字段在域上的共差就是该操作员的一个例子。该估计值是集成操作员及其平根的Hilbert-Schmidt规范形式,由拥有Drichlet或Neumann类型等同边界条件的椭圆操作员的分数功率构成。这些估计值对封闭域的随机局部偏差方程式及其数字近似具有重要影响,它们将驱动噪音的规律性与差异操作员的特性结合起来。用于得出估计的主要工具是复制Hilbert内核域的功能空间以及Sobolev空间的Hilbert-Schmidt嵌入的特性。两种无血源和同质的内核都得到了考虑。本文所覆盖的同质内核的重要实例包括了Mat\'ern内核的类别。