On anisotropic Triebel‐Lizorkin type spaces, with applications to the study of pseudo‐differential operators

Abstract

A construction of Triebel-Lizorkin type spaces associated with flexible decompositions of the frequency spaceℝdis considered. The class of admissible frequency decompositions is generated by a one parameter group of (anisotropic) dilations onℝdand 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.

Anisotropic classes of homogeneous pseudodifferential symbols

“…Before stating our result, we recall the definitions and the molecular characterizations of these spaces. For an extensive treatment of these spaces, the reader is referred to [4,5,6]; see also [2].…”

Compactly Supported Frames for Decomposition Spaces

“…To construct the resolution of the identity, we use a suitable covering of the frequency space. For a much more detailed discussion of the T-L type spaces see [2], and for the associated modulation spaces see [1].…”

“…It can be shown that F s p,q (h) depends only on h up to equivalence of the norms (see [2,Proposition 5.3]), so the T-L type spaces are well-defined and similar for the modulation spaces. Furthermore, they both constitute quasi-Banach spaces, and for p, q < ∞, S is dense in both (see [2,Proposition 5.2]).…”

“…Frames for T-L type spaces and the associated modulation spaces have been considered earlier: Banach frames for α-modulation spaces in [4,9], and Banach frames for T-L type spaces and the associated modulation spaces where constructed in [1,2]. However, these frames were constructed using bandlimited functions which rules out compact support in direct space.…”

“…With this perturbation principle, finite linear combinations of shifts and dilates of a single functions with sufficient decay in direct and frequency space can be used to construct frame expansions with a prescribed nature such as compact support. These frame expansions are constructed from the already known Banach frames in [1,2]; thereby, generating frame expansions which share the same sparseness properties as the already known representations.…”