Matrix-Weighted Besov-Type and Triebel-Lizorkin-Type Spaces

Fan Bu, Tuomas P. Hytönen, Dachun Yang, Wen Yuan

Published 2023-04-01Version 1

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 memoir, we introduce and study 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 $\mathbb R^n$. Our contributions 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 Calderon-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.