Matrix-weighted Besov-type and Triebel–Lizorkin-type spaces

Besov-type and Triebel–Lizorkin-type spaces $\dot B^{s,\tau}_{p,q}$ and $\dot F^{s,\tau}_{p,q}$ on $\mathbb{R}^n$ consist of a general family of function spaces that cover not only the well-known Besov and Triebel–Lizorkin spaces $\dot B^{s}_{p,q}$ and $\dot F^{s}_{p,q}$ (when $\tau=0$) but also several other spaces of interest, such as Morrey spaces and $Q$ spaces. In this talk, we introduce matrix-weighted versions $\dot B^{s,\tau}_{p,q}(W)$ and $\dot F^{s,\tau}_{p,q}(W)$ of these general function spaces on $\mathbb{R}^n$, where $W$ is a matrix-valued Muckenhoupt $A_p$ weight on $\mathbb R^n$. The main contents include several characterizations of these spaces in terms of both the $\varphi$-transform of Frazier and Jawerth and the related sequence spaces $\dot b^{s,\tau}_{p,q}(W)$ and $\dot f^{s,\tau}_{p,q}(W)$, almost diagonal conditions that imply the boundedness of weakly defined operators on these spaces, and consequences for the boundedness of classical operators like pseudo-differential operators, trace operators, and Calderón–Zygmund operators. Results of this type are completely new on this level of generality, but many of them also improve the known results in the unweighted spaces $\dot B^{s,\tau}_{p,q}$ and $\dot F^{s,\tau}_{p,q}$ or, with $\tau=0$, in the weighted spaces $\dot B^{s}_{p,q}(W)$ and $\dot F^{s}_{p,q}(W)$. Several of our results are conveniently stated in terms of a new concept of the $A_p$-dimension $d\in[0,n)$ of a matrix weight $W\in A_p$ on $\mathbb R^n$ and, in several cases, the obtained estimates are shown to be sharp. In particular, for certain parameter ranges, we are able to characterize the sharp almost diagonal conditions that imply the boundedness of operators on these spaces.
This talk is based on the recent joint works with Fan Bu, Tuomas P. Hytönen and Wen Yuan.