Let F be a distribution function. Its characteristic function belongs to ${\operatorname {Lip}}\alpha ,0 < \alpha < 1$, if and only if $F( - x)$ and $1 - F(x)$ are $O(x^{ - \alpha } )$ as $x \to \infty $ (see Boas [1]). The n-dimensional Fourier transform of a radial function reduces to the Hankel transform of a function in one variable. Results similar to those given by Boas are obtained for this transform. The problem, however, is discussed in a rather general form. The class of functions $\Phi $, $\Phi (x) = \int _0^\infty k(xt)dF(t)$, is considered. It is assumed that k is essentially bounded and has a nonzero Peano derivative of some definite order at zero, whereas $F(t)$ is nonincreasing but not necessarily bounded.