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Abstract. Weighted anisotropic Triebel-Lizorkin spaces are introduced and studied with the use of discrete wavelet transforms. This study extends the isotropic methods of dyadic ϕ-transforms of Jawerth (1985, 1989) to non-isotropic settings associated with general expansive matrix dilations and A ∞ weights.In close analogy with the isotropic theory, we show that weighted anisotropic Triebel-Lizorkin spaces are characterized by the magnitude of the ϕ-transforms in appropriate sequence spaces. We also introduce non-isotropic analogues of the class of almost diagonal operators and we obtain atomic and molecular decompositions of these spaces, thus extending isotropic results of Frazier and Jawerth.

Abstract. A criteria on the vector-valued Banach function spaces X (B) is obtained so that whenever a vector-valued singular integral operator is bounded on X (B), it can be extended to be a bounded linear operator on the corresponding Morrey type spaces. Using this result, we define the generalized Triebel-Lizorkin-Morrey spaces and obtain the atomic and molecular decompositions. As a particular example of the generalized Triebel-Lizorkin-Morrey spaces, we introduce and study the variable Triebel-Lizorkin-Morrey spaces.

The Hardy-Morrey spaces with variable exponents are introduced in terms of maximal functions. The atomic decomposition of Hardy-Morrey spaces with variable exponents is established. This decomposition extends and unifies several atomic decompositions of Hardy type spaces such as the Hardy-Morrey spaces and the Hardy spaces with variable exponents. Some applications of this atomic decomposition on singular integral are presented.