Atomic decomposition for weighted Besov and Triebel-Lizorkin spaces

Rychkov defined weighted Besov spaces and weighted Triebel-Lizorkin spaces coming with a weight in the class \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$A_p^{\rm loc}$\end{document}, which is even wider than the class A_p due to Muckenhoupt. In the present paper we are concerned with the atomic decomposition of these spaces. As an application, we obtain some key theorems in the theory of function spaces. Finally we conclude this paper with some helpful examples showing the difference between weighted spaces and unweighted spaces.