Functional means, convolution operators and semigroups

The generalizations of Levy's functional means are considered, which are the limits of integral means over the infinitely-divisible product-measures. Convolution operators with the family of such means form the C-0-semigroup generated by the non-Gaussian generalization of the Levy-Laplacian. The concentration of means near the sphere of a certain radius is verified. Behaviour of this radius in time is studied and its relation to the analytical properties of the operator semigroup. By way of application the action of convolutions on the finite-supported functions is described.