This paper studies the weighted Sobolev spaces \(L^{p,s}(\omega) := L^{p,s}(R^n, \omega(x)dx)\), \((1 < p < \infty, s \in N)\), where \(\omega\) belongs to one of the Muckenhoupt classes \(A_p\) or \(A_p^{loc}\).
The paper is organized as follows: Section 1 is introductory. Section 2 consists of preliminaries: the fundamental theory of wavelets is described, as well as the classes \(A_p\) and \(A_p^{loc}\), some bases, weighted function spaces, and some known results on \(L^p(\omega)\). A characterization of \(L^{p,s}(\omega)\) with \(\omega \in A_p\) by wavelets is given in Section 3, whereas Section 4 presents a characterization of \(L^{p,s}(\omega)\) with \(\omega \in A_p^{loc}\), in terms of wavelets and scaling functions. Finally, in Section 5, the unconditional bases and greedy bases in \(L^{p,s}(\omega)\) are constructed, using the results of the two previous sections.