On Anisotropic Triebel-lizorkin Type Spaces, with Applications to the Study of Pseudo-differential Operators

A construction of Triebel-Lizorkin type spaces associated with flexible decompositions of the frequency space R is considered. The class of admissible frequency decompositions is generated by a one parameter group of (anisotropic) dilations on R and a suitable decomposition function. The decomposition function governs the structure of the decomposition of the frequency space, and for a very particular choice of decomposition function the spaces are reduced to classical (anisotropic) Triebel-Lizorkin spaces. An explicit atomic decomposition of the Triebel-Lizorkin type spaces is provided, and their interpolation properties are studied. As the main application, we consider Hörmander type classes of pseudo-differential operators adapted to the anisotropy and boundedness of such operators between corresponding Triebel-Lizorkin type spaces is proved.