Bourgain-Morrey-Lorentz spaces and operators on them
Tengfei Bai, Pengfei Guo, Jingshi Xu
TL;DR
The article defines and analyzes Bourgain-Morrey-Lorentz spaces $M_{p,q}^{t,r} ({{{ R}}^n})$, establishing their predual via block spaces $\,\mathcal{H}_{ p', q'}^{t', r'} ({{{ R}}^n})$ and proving a full duality framework. Employing this structure, it proves boundedness results for the Hardy-Littlewood maximal operator, sharp maximal operator, Calderón-Zygmund operators, fractional integrals, and their commutators on these spaces, and develops a weak Hardy factorization leading to BMO characterizations via commutators with homogeneous CZ operators. A Hardy factorization theorem and the Köthe-dual framework underpin a characterization of BMO, while a sharp compactness result shows that commutators are compact on $M_{p,q}^{t,r}$ precisely when the symbol lies in $\operatorname{CMO}$. Overall, the work extends harmonic analysis tools to a broad Morrey-Lorentz-type setting, with implications for PDEs and related function spaces.
Abstract
We introduce Bourgain-Morrey-Lorentz spaces and give a description of the predual of Bourgain-Morrey-Lorentz spaces via the block spaces. As an application of duality, we obtain the boundedness of Hardy-Littlewood maximal operator, sharp maximal operator, Calderón-Zygmund operator, fractional integral operator, commutator on Bourgain-Morrey-Lorentz spaces. Moreover, we obtain a weak Hardy factorization terms of Calderón-Zygmund operator in Bourgain-Morrey-Lorentz spaces. Using this result, we obtain a characterization of functions in $\BMO$ (the functions of ``bounded mean oscillation'') via the boundedness of commutators generated by them and a homogeneous Calderón-Zygmund operator. In the last, we show that the commutator generated by a function $b$ and a homogeneous Calderón-Zygmund operator is a compact operator on Bourgain-Morrey-Lorentz spaces if and only if $b$ is the limit of compactly supported smooth functions in $\BMO$.