Classification of anisotropic Triebel-Lizorkin spaces

Area of Science:

• Mathematics
• Harmonic Analysis
• Functional Analysis

Background:

• Anisotropic homogeneous Triebel-Lizorkin spaces are crucial in harmonic analysis.
• Characterizing these spaces aids in understanding their properties and applications.
• Previous work classified anisotropic Hardy spaces.

Purpose of the Study:

• To characterize expansive matrices generating the same anisotropic homogeneous Triebel-Lizorkin space.
• To establish conditions for matrix equivalence in generating these spaces.
• To extend existing classifications of anisotropic function spaces.

Main Methods:

• Utilizing the theory of anisotropic homogeneous Triebel-Lizorkin spaces.
• Analyzing properties of expansive matrices.
• Comparing homogeneous quasi-norms associated with matrices.

Main Results:

• Established that matrices A and B generate the same anisotropic homogeneous Triebel-Lizorkin space if and only if their associated homogeneous quasi-norms are equivalent.
• Identified an exception for the case $n=m$ with $A=B$.

Conclusions:

• The characterization provides a deeper understanding of anisotropic homogeneous Triebel-Lizorkin spaces.
• Results complement and extend the classification of anisotropic Hardy spaces.
• This work contributes to the broader study of function spaces and matrix properties.