Trace and extension operators for Besov spaces and Triebel–Lizorkin spaces with variable exponents

This paper is concerned with the boundedness of trace and extension operators for Besov spaces and Triebel–Lizorkin spaces with variable exponents on the upper half space R+n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb R}^n_+$$\end{document}. To define trace and extension operators, we introduce a quarkonial decomposition for Besov spaces and Triebel–Lizorkin spaces with variable exponents on Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb R}^n$$\end{document}. We then study the continuity of such operators related to R+n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb R}^n_+$$\end{document}.