Equivalent norms and $\phi$-transform of matrix-weighted anisotropic Besov-type and Triebel-Lizorkin-type spaces

We introduce matrix-weighted anisotropic Besov-type and Triebel-Lizorkin-type spaces associated with an expansive matrix $A$. Inspired by the ${\text{A}}_p$-dimensions of matrix weight of Bu et al. (2025), we study the properties of matrix weight associated with $A$. Using the nice properties of matrix weight, we obtain that these spaces are equivalent with their corresponding averaging spaces and establish the discrete $\phi$-transform of these spaces. Finally, we introduce matrix-weighted anisotropic Triebel-Lizorkin spaces for the limiting case $p=\mathrm{\infty }$ and obtain their $\phi$-transform. The relation between matrix-weighted anisotropic Triebel-Lizorkin spaces and matrix-weighted anisotropic Besov-type and Triebel-Lizorkin-type spaces is also studied.