Littlewood-Paley theory and function spaces with A(p)(loc) weights

Characterizations via convolutions with smooth compactly supported kernels and other distinguished properties of the weighted Besov-Lipschitz and Trie bel-Lizorkin spaces on R-n with weights that are locally in A(p) but may gr ow or decrease exponentially at infinity are investigated. Square-function characterizations of the weighted L-P and Hardy spaces with the above class of weights are also obtained. A certain local variant of the Calderon repr oducing formula is constructed and widely used in the proofs.