Estimates of the uniform modulus of continuity for Bessel potentials

In the theory of function spaces it is an important problem to describe the differential properties for the classical Bessel and Riesz potentials as well as for their generalizations. Bessel potentials are determined by the convolutions of functions with Bessel-MacDonald kernels G α. In this paper we characterize the integral properties of functions by their decreasing rearrangements. The differential properties of potentials are characterized by their modulus of continuity of order k in the uniform norm. Estimates of such type were obtained by A. Gogatishvili, J. Neves, and B. Opic in the case k > α. Here, we remove this restriction and obtain the results for all values k ε N. We find order-sharp estimates from above for moduli of continuity and construct the examples confirming the sharpness. On the base of these results we obtain the order-sharp estimates for continuity envelope function in the space of potentials, and give estimates for the approximation numbers of the embedding operator. © 2013 Pleiades Publishing, Ltd.