On the mixed Bourgain-Morrey spaces
Tengfei Bai, Pengfei Guo, Jingshi Xu
TL;DR
The paper introduces mixed Bourgain-Morrey spaces $M_{\vec{p}}^{t,r}$ and their preduals $\mathcal{H}_{\vec{p}'}^{t',r'}$, establishing a full harmonic-analysis framework: duality, density and approximation, and reflexivity. It proves the boundedness of the Hardy-Littlewood maximal operator, the iterated maximal operator, the fractional integral operator $I_\alpha$, and singular integrals on these spaces, and develops Littlewood-Paley theory, heat-sequence characterizations, and wavelet descriptions. Block spaces $\mathcal{H}_{\vec{p}'}^{t',r'}$ are shown to be the preduals of $M_{\vec{p}}^{t,r}$, with a robust structure including completeness, Fatou and lattice properties, and associative duality. Applications include wavelet characterizations and a fractional chain rule within mixed Bourgain-Morrey Triebel-Lizorkin spaces, highlighting both the depth and versatility of the mixed-norm Bourgain-Morrey framework.
Abstract
We introduce the mixed Bourgain-Morrey spaces and obtain their preduals. The boundedness of Hardy-Littlewood maximal operator, iterated maximal operator, fractional integral operator, singular integral operator on these spaces is proved. The Littlewood-Paley theory for mixed Bourgain-Morrey spaces and their preduals are established. As applications, we consider wavelet characterizations for mixed Bourgain-Morrey spaces and a fractional chain rule in mixed Bourgain-Morrey Triebel-Lizorkin spaces. In addition, we give a description of the dual of mixed Bourgain-Morrey spaces and conclude the reflexivity of these spaces.