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

[摘要] 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” $\mathbf{D}_{\varepsilon }^{\alpha }f$ is introduced, that is generated by the modified Poisson semigroup and associated with the Flett potentials ( $0<\alpha <\infty $ , $\varphi \in L_{p}(\mathbb{R}^{n})$ ). Then the relationship between the order of “ $L_{p}$ -smoothness” of a function f and the “rate of $L_{p}$ -convergence” of the families $\mathbf{D}_{\varepsilon }^{\alpha } \mathcal{F}^{\alpha }f$ to the function f as $\varepsilon \rightarrow 0^{+}$ is also obtained.

[学科分类] 电力

[关键词] Truncated hypersingular integrals;Flett potentials;Poisson semigroup;Rate of convergence