Classes of kernels and continuity properties of the tangential gradient of an integral operator in Hölder spaces on a manifold

We prove multiplication and embedding theorems for classes of kernels of integral operators in subsets of metric spaces with a measure. Then we prove a tangential differentiation theorem with respect to a semi-tangent vector for integral operators that are defined on an upper-Ahlfors regular subset of the Euclidean space and a continuity theorem for the corresponding integral operator in Hölder spaces in the specific case of a differentiable manifold.