Ascent and descent of multiplication and composition induced operators on variable exponent lebesgue spaces

The paper focuses on determining the ascent and descent of multiplication, composition, and weighted composition operators on variable exponent Lebesgue spaces. We explore the conditions on the measurable functions u and measurable transformations T defined on sigma-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma -$$\end{document}finite complete measure space (X,A,mu)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X,\mathcal {A},\mu )$$\end{document} that cause these operators on variable exponent Lebesgue spaces to have finite or infinite ascent (descent).