Dominated Compactness Theorem in Banach Function Spaces and its Applications

A famous dominated compactness theorem due to Krasnosel’skiĭ states that compactness of a regular linear integral operator in Lp follows from that of a majorant operator. This theorem is extended to the case of the spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p(\cdot)}(\Omega, \mu, \varrho), \mu \Omega < \infty$$\end{document}, with variable exponent p(·), where we also admit power type weights \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varrho$$\end{document}. This extension is obtained as a corollary to a more general similar dominated compactness theorem for arbitrary Banach function spaces for which the dual and associate spaces coincide. The result on compactness in the spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p(\cdot)}(\Omega, \mu, \varrho)$$\end{document} is applied to fractional integral operators over bounded open sets.