On matrix weighted Bourgain-Morrey Triebel-Lizorkin spaces
Tengfei Bai, Pengfei Guo, Jingshi Xu
TL;DR
This work extends harmonic analysis to matrix-weighted Morrey-type scales by introducing homogeneous and inhomogeneous matrix-weighted Bourgain-Morrey Triebel-Lizorkin spaces $\dot{F}_{p,t,r}^{s,q}(W)$ and $F_{p,t,r}^{s,q}(W)$. It establishes robust norm equivalences across several realizations via $A_Q$-reductions, and develops comprehensive characterizations through Peetre-type maximal functions, Lusin-area, Littlewood-Paley $g_{\lambda}^{*}$, wavelets, atoms, and approximation methods. The results yield boundedness of Calderón-Zygmund and pseudo-differential operators with symbols in Hörmander classes and Hölder-Zygmund classes on these matrix-weighted spaces, significantly broadening the operator theory in matrix-weighted Triebel-Lizorkin-Morrey contexts. Overall, the paper provides a unified, technically detailed framework for matrix-weighted Morrey-type function spaces and their applications to pseudodifferential analysis.
Abstract
We introduce the homogeneous (inhomogeneous) matrix weighted Bourgain-Morrey Triebel-Lizorkin spaces and obtain their equivalent norms. We also obtain their characterizations by Peetre type maximal functions, Lusin-area function, Littlewood-Paley $g_λ^{*}$-function, approximation, wavelet and atom. As an application, we obtain boundedness of pseudo-differential operators with symbols in the Hörmander classes and Hölder-Zygmund classes on inhomogeneous matrix weighted Bourgain-Morrey Triebel-Lizorkin spaces.