标题： 矩阵加权Besov-Triebel-Lizorkin空间的最佳刻度：实变特征刻画，可积指数的不变性及Sobolev型嵌入 显示英文标题

标题： Matrix-Weighted Besov-Triebel-Lizorkin Spaces of Optimal Scale: Real-Variable Characterizations, Invariance on Integrable Index, and Sobolev-Type Embedding

摘要： 本文中，利用增长函数我们引入了带有矩阵 $\mathcal{A}_{\infty}$ 权重的广义矩阵加权Besov-Triebel-Lizorkin型空间。 我们首先分别通过 $\varphi$ 变换、Peetre型极大函数和Littlewood-Paley函数来刻画这些空间。 此外，在建立相应序列空间上几乎对角算子的有界性之后，我们得到了这些空间的分子和小波特征化。 作为应用，我们找到了Triebel-Lizorkin型空间在可积指数下的不变性的充分必要条件，以及所有这些空间的Sobolev型嵌入的充分必要条件。 主要的新颖之处在于这些结果具有广泛的普适性，增长函数的增长条件不仅对于几乎对角算子的有界性是充分的，而且也是必要的，因此这种新的Besov-Triebel-Lizorkin型框架是最优的。某些结果要么是全新的，要么即使对于已知的矩阵加权Besov-Triebel-Lizorkin空间也有所改进，而且更进一步地，即使在标量值设定下，所有结果也都是新的。 显示英文摘要

摘要： In this article, using growth functions we introduce generalized matrix-weighted Besov-Triebel-Lizorkin-type spaces with matrix $\mathcal{A}_{\infty}$ weights. We first characterize these spaces, respectively, in terms of the $\varphi$-transform, the Peetre-type maximal function, and the Littlewood-Paley functions. Furthermore, after establishing the boundedness of almost diagonal operators on the corresponding sequence spaces, we obtain the molecular and the wavelet characterizations of these spaces. As applications, we find the sufficient and necessary conditions for the invariance of those Triebel-Lizorkin-type spaces on the integrable index and also for the Sobolev-type embedding of all these spaces. The main novelty exists in that these results are of wide generality, the growth condition of growth functions is not only sufficient but also necessary for the boundedness of almost diagonal operators and hence this new framework of Besov-Triebel-Lizorkin-type is optimal, some results either are new or improve the known ones even for known matrix-weighted Besov-Triebel-Lizorkin spaces, and, furthermore, even in the scalar-valued setting, all the results are also new.