Atomic and molecular decompositions of anisotropic Besov spaces

In this work we develop the theory of weighted anisotropic Besov spaces associated with general expansive matrix dilations and doubling measures with the use of discrete wavelet transforms. This study extends the isotropic Littlewood- Paley methods of dyadic phi-transforms of Frazier and Jawerth [19, 21] to non-isotropic settings. Several results of isotropic theory of Besov spaces are recovered for weighted anisotropic Besov spaces. We show that these spaces are characterized by the magnitude of the phi-transforms in appropriate sequence spaces. We also prove boundedness of an anisotropic analogue of the class of almost diagonal operators and we obtain atomic and molecular decompositions of weighted anisotropic Besov spaces, thus extending isotropic results of Frazier and Jawerth [21].