Estimation of the uniform modulus of continuity of the generalized Bessel potential

We study generalized Bessel potentials constructed by means of convolutions of functions with kernels that generalize the classical Bessel–Macdonald kernels. In contrast to the classical case, nonpower singularities of kernels in a neighborhood of the origin are admitted. The integral properties of functions are characterized in terms of decreasing rearrangements. The differential properties of potentials are described by the $k$th-order moduli of continuity in the uniform norm. An order-sharp upper estimate is established for the modulus of continuity of a potential. Such estimates play an important role in the theory of function spaces. They allow one to establish sharp embedding theorems for potentials, find majorants of the moduli of continuity, and estimate the approximation numbers of embedding operators.