Estimates for continuity envelopes and approximation numbers of Bessel potentials

In this paper we study spaces of Bessel potentials in nn-dimensional Euclidean spaces. They are constructed on the basis of a rearrangement-invariant space (RIS) by using convolutions with Bessel–MacDonald kernels. Specifically, the treatment covers spaces of classical Bessel potentials. We establish two-sided estimates for the corresponding modulus of smoothness of order k∈Nk∈N, ωk(f;t)ωk(f;t), and determine their continuity envelope functions. This result is then applied to estimate the approximation numbers of some embeddings.