Characterization of a class of embeddings for function spaces with Muckenhoupt weights

For function spaces equipped with Muckenhoupt weights, the validity of continuous Sobolev embeddings in case p0≤p1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${p_0 \leq p_1}$$\end{document} is characterized. Extensions to Jawerth–Franke embeddings, vector-valued spaces, and examples involving some prominent weights are also provided.