Аннотация:
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.