THE CHARACTERIZATIONS OF WEIGHTED SOBOLEV SPACES BY WAVELETS AND SCALING FUNCTIONS

Abstract

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 $A_p^{\mathop{\mathrm{loc}}}$ weights. In the case of $w \in A_p$, we use only wavelets with proper regularity. Meanwhile, if we assume $w \in A_p^{\mathop{\mathrm{loc}}}$, 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.