The rate of convergence of truncated hypersingular integrals generated by the Poisson and metaharmonic semigroups

In harmonic analysis, an important problem is to obtain inversion formulas for the potential-type integral operators. The studies on this subject have been developed by the use of hypersingular integral technique. In this paper the families of truncated hypersingular integrals generated by the Poisson and metaharmonic semigroups and dependent on a parameter epsilon, are introduced. Then the connection between the order of smoothness of a given L-p-function phi and the rate of convergence of these families of truncated hypersingular integrals, which converge to phi when epsilon tends to 0, is obtained.