The characterizations of weighted sobolev spaces by wavelets and scaling functions

We prove that suitable wavelets and scaling functions give characterizations and unconditional bases of the weighted Sobolev space L^p,s(w) with A_p or A^loc_p weights. In the case of w ∈ A_p, we use only wavelets with proper regularity. Meanwhile, if we assume w ∈ A^loc_p, not only compactly supported C^s+1-wavelets but also compactly supported C^s+1-scaling functions come into play. We also establish that our bases are greedy for L^p,s(w) after normalization.