Essential norm of the extensions of Stevic-Sharma operator on some spaces of analytic functions
Mostafa Hassanlou, Hussain Gissy
TL;DR
The paper addresses the problem of estimating the essential norms of two extensions of the Stevic-Sharma operator, $T_{\psi_1,\psi_2,\varphi}^n$ and $T_{\psi_1,\psi_2,\varphi}^{m,n}$, acting between $\mathcal{Q}_K(p,q)$ or $H^{\infty}$ and the weighted Bloch space $\mathcal{B}_{\mu}$. It develops exact order estimates by leveraging boundary-growth functionals $A(u,\varphi,\gamma)$ and a test-function framework, providing both upper and lower bounds that coincide in key cases. The main contributions are explicit formulas for the essential norms: $\|T_{\psi_1,\psi_2,\varphi}^n\|_{e}$ in terms of $A(\cdot,\cdot,\cdot)$ and $\gamma=(q+2)/p$, and $\|T_{\psi_1,\psi_2,\varphi}^{m,n}\|_{e}$ in terms of $E_i$ boundary quantities for $i\in\{m,m+1,n,n+1\}$, including a refined result when $m+1=n$. These findings extend prior work on compactness and essential norms for related operator classes on these spaces, with implications for operator stability and analytic-function space theory.
Abstract
In this paper, we consider two extensions of Stevic-Sharma operator and find estimations for the essential norm of them from QK(p; q) and H1 into weighted Bloch spaces.