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.

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修正的Poisson半群生成的截断的超奇异积分的收敛速度

超奇异积分已经成为有效的工具，可以逆转多维势能型算子，例如Riesz，Bessel，Flett，抛物线势等。它们（至少正式地）代表了合适的微分算子的分数次幂。本文介绍了所谓的“截断的超奇异积分算子” $ \ mathbf {D} _ {\ varepsilon} ^ {\ alpha} f $的族，该族由修改后的Poisson半群生成并与Flett相关电位（$ 0 <\ alpha <\ infty $，$ \ varphi \ in L_ {p}（\ mathbb {R} ^ {n}）$）。然后，函数f的“ $ L_ {p} $-平滑度”的顺序与家庭$ \ mathbf {D} _ {\ varepsilon} ^的“ $ L_ {p} $-收敛的速率”之间的关系还获得函数f的{\ alpha} \ mathcal {F} ^ {\ alpha} f $作为$ \ varepsilon \ rightarrow 0 ^ {+} $。