On the approximation properties of bi-parametric potential-type integral operators

Abstract

In this work we study the approximation properties of the classical Riesz potentials $I^{\alpha }f\equiv (-\Delta )^{-\alpha /2}f$ and the so-called bi-parametric potential-type operators $J_{\beta }^{\alpha }f\equiv(E+(-\Delta )^{\beta /2})^{-\alpha /\beta }f$ as $\alpha \rightarrow \alpha_{0}>0$ where, $\alpha >0$, $\beta >0$, $E$ is the identity operator and $\Delta $ is the laplacian. These potential-type operators generalize the famous Bessel potentials when $\beta =2$ and Flett potentials when $\beta =1$. We show that, if $A^{\alpha}$ is one of operators $J_{\beta }^{\alpha }$ or $I^{\alpha}$, then at every Lebesgue point of $f\in L_{p}(\mathbb{R}^{n})$ the asymptotic equality $(A^{\alpha}f)(x)-(A^{\alpha _{0}}f)(x)=O(1)(\alpha-\alpha _{0})$, ($\alpha \rightarrow \alpha _{0}^{+}$) holds. Also the asymptotic equality $\left\Vert A^{\alpha }f-A^{\alpha _{0}}f\right\Vert_{p}=O(1)(\alpha -\alpha _{0})$, ($\alpha \rightarrow \alpha _{0}^{+}$) holds when $A^{\alpha}=J_{\beta }^{\alpha }$.

Keywords

• Abel-Poisson semigroup
• Gauss-Weierstrass semigroup
• Riesz potentials
• Bessel potentials
• potentials-type operators

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