One-sided invertibility of binomial functional operators with a shift on rearrangement-invariant spaces

Let Γ be an oriented Jordan smooth curve and α a diffeomorphism of Γ onto itself which has an arbitrary nonempty set of periodic points. We prove criteria for one-sided invertibility of the binomial functional operator\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$A = aI - bW$$ \end{document} wherea andb are continuous functions,I is the identity operator,W is the shift operator,Wf=foα, on a reflexive rearrangement-invariant spaceX(Γ) with Boyd indices αX , βXand Zippin indicespx,qx satisfying inequalities\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$0< \alpha x = px \leqslant qx = \beta x< 1$$ \end{document}