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

Abstract

Abstract 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 $\mathbf{D}_{\varepsilon }^{\alpha }f$ 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 < ∞ $0<\alpha <\infty $ , φ ∈ L p ( R n ) $\varphi \in L_{p}(\mathbb{R}^{n})$ ). 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 D ε α F α f $\mathbf{D}_{\varepsilon }^{\alpha } \mathcal{F}^{\alpha }f$ to the function f as ε → 0 + $\varepsilon \rightarrow 0^{+}$ is also obtained.

Keywords

• Truncated hypersingular integrals
• Flett potentials
• Poisson semigroup
• Rate of convergence