Essential norm of generalized integral type operator from $\mathcal{Q}_{K}(p,q)$ to Zygmund Spaces

Abstract

‎Let $\varphi$ be an analytic self-map on $\mathbb{D}$‎, ‎$n\in\mathbb{N}$ and $g\in H(\mathbb{D})$‎. ‎We consider the essential norm of the generalized integral-type operator $C_{\varphi,g}^n:\mathcal{Q}_K \left(p,q\right)\to {\mathcal Z}_{\mu }$ that is defined as follows‎

‎$$\left(C_{\varphi,g}^nf\right)(z)=\int_{0}^{z}f^{(n)}(\varphi(\xi))g(\xi)\ d\xi,$$‎

‎for all $f\in \mathcal{Q}_K \left(p,q\right)$‎. ‎We give an estimate for the essential norm of the above operator‎.