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

Abstract

AbstractHypersingular 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” $\mathbf{D}_{\varepsilon }^{\alpha }f$ 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<\alpha <\infty $ 0 < α < ∞ , $\varphi \in L_{p}(\mathbb{R}^{n})$ φ ∈ L p ( R n ) ). Then the relationship between the order of “$L_{p}$ L p -smoothness” of a function f and the “rate of $L_{p}$ L p -convergence” of the families $\mathbf{D}_{\varepsilon }^{\alpha } \mathcal{F}^{\alpha }f$ D ε α F α f to the function f as $\varepsilon \rightarrow 0^{+}$ ε → 0 + is also obtained.