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

Authors : Çağla SEKİN, Mutlu GÜLOĞLU, İlham ALİYEV

Pages : 1681-1691

Doi:10.15672/hujms.821159

View : 23 | Download : 8

Publication Date : 2021-12-14

Article Type : Research Paper

Abstract :In this work we study the approximation properties of the classical Riesz potentials $I^{\alpha }f\equiv insert ignore into journalissuearticles values(-\Delta );^{-\alpha /2}f$ and the so-called bi-parametric potential-type operators $J_{\beta }^{\alpha }f\equivinsert ignore into journalissuearticles values(E+insert ignore into journalissuearticles values(-\Delta );^{\beta /2});^{-\alpha /\beta }f$ 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 $f\in L_{p}insert ignore into journalissuearticles values(\mathbb{R}^{n});$ the asymptotic equality $insert ignore into journalissuearticles values(A^{\alpha}f);insert ignore into journalissuearticles values(x);-insert ignore into journalissuearticles values(A^{\alpha _{0}}f);insert ignore into journalissuearticles values(x);=Oinsert ignore into journalissuearticles values(1);insert ignore into journalissuearticles values(\alpha-\alpha _{0});$, insert ignore into journalissuearticles values($\alpha \rightarrow \alpha _{0}^{+}$); holds. Also the asymptotic equality $\left\Vert A^{\alpha }f-A^{\alpha _{0}}f\right\Vert_{p}=Oinsert ignore into journalissuearticles values(1);insert ignore into journalissuearticles values(\alpha -\alpha _{0});$, insert ignore into journalissuearticles values($\alpha \rightarrow \alpha _{0}^{+}$); holds when $A^{\alpha}=J_{\beta }^{\alpha }$.
Keywords : Abel Poisson semigroup, Gauss Weierstrass semigroup, Riesz potentials, Bessel potentials, potentials type operators