The main purpose of this paper is to extend the isotropic methods of dyadic -transforms of M. Frazier and B. Jawerth in [Indiana Univ. Math. J. 34, 777–799 (1985; Zbl 0551.46018); J. Funct. Anal. 93, No. 1, 34–170 (1990; Zbl 0716.46031)] to non-isotropic settings associated with weights and general expansive matrix dilations which are real matrices all of whose eigenvalues satisfy . Such expansive matrix dilations have fair generality and wide applications in the multidimensional theory of wavelets, which are the main advantage and novelty of this approach.
Let be a such dilation and the adjoint (transpose) of . Let be a Schwartz function on satisfying and for all , where denotes the Fourier transform of . For , , and (the Muckenhoupt class associated to ), the authors define the weighted anisotropic Triebel-Lizorkin space as the collection of all tempered distributions (modulo polynomials) such that
In the standard dyadic case , and then factor in the above definition would be instead of the classical . Using the Calderón reproducing formula, the authors prove that the definition of is independent of the choice of .
Moreover, in close analogy with the isotropic theory, the authors show that these weighted anisotropic Triebel-Lizorkin spaces are characterized by the magnitude of the -transforms in appropriate sequence spaces. The authors also introduce non-isotropic analogues of the class of almost diagonal operators and obtain atomic and molecular decompositions of these spaces, thus extending the isotropic result of Frazier and Jawerth mentioned above. Finally, the authors also study inhomogeneous weighted anisotropic Triebel-Lizorkin spaces and outline analogous decomposition results for these spaces.