Besov Spaces with General Weights

We introduce Besov spaces with general smoothness. These spaces unify and generalize the classical Besov spaces. We establish the phi-transform characteriza-tion of these spaces in the sense of Frazier and Jawerth and we prove their Sobolev embeddings. We establish the smooth atomic, molecular and wavelet decomposition of these function spaces. A characterization of these function spaces in terms of the difference relations is given.