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 $A^{\alpha}=J_{\beta }^{\alpha }$.
📌 3 Maddede Bu Makale
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Amaç & Hipotez: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.
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Temel Bulgular & Veri:These potential-type operators generalize the famous Bessel potentials when $\beta =2$ and Flett potentials when $\beta =1$.
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Sonuç & Katkı: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 $A^{\alpha}=J_{\beta }^{\alpha }$.
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 $A^{\alpha}=J_{\beta }^{\alpha }$.
Aliev, İ. A., Sekin, Ç., Güloğlu, M. (2022). On the approximation properties of bi-parametric potential-type integral operators. Hacettepe Journal of Mathematics and Statistics, Cilt: 50 - Sayı: 6, 1681–1691. https://ajindex.com/articles/529474
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