The convergence rate of truncated hypersingular integrals generated by the modified Poisson semigroup.

Hypersingular integrals have appeared as effective tools for inversion of multidimensional potential-type operators such as Riesz, Bessel, Flett, parabolic potentials, etc. They represent (at least formally) fractional powers of suitable differential operators. In this paper the family of the so-called "truncated hypersingular integral operators" D ε α f is introduced, that is generated by the modified Poisson semigroup and associated with the Flett potentials F α φ = (E + − Δ) − α φ (0 < α < ∞ , φ ∈ L p (R n) ). Then the relationship between the order of " L p -smoothness" of a function f and the "rate of L p -convergence" of the families D ε α F α f to the function f as ε → 0 + is also obtained. [ABSTRACT FROM AUTHOR]