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

TL;DR

This paper introduces a broad class of matrix-weighted Besov-Triebel-Lizorkin spaces using growth functions, providing characterizations, invariance conditions, and embeddings, with results that are optimal, general, and extend existing theories.

Contribution

It develops a new, optimal framework for matrix-weighted Besov-Triebel-Lizorkin spaces with comprehensive characterizations and invariance results, extending and improving prior work.

Findings

Characterization via $ ext{phi}$-transform, maximal, and Littlewood-Paley functions

Boundedness of almost diagonal operators established

Necessary and sufficient conditions for invariance and embeddings

Abstract

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.