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arXiv 2602.00322math.FA
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Bourgain-Morrey序列空间：结构性质、与经典ℓ^p空间的关系及对偶性

Bourgain-Morrey sequence spaces: structural properties, relations to classical $\ell^{p}$ spaces and duality

Universidad de Sonora（索诺拉大学）

Francisco Alejandro Villegas Acuña

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AI总结：

本文研究了离散Bourgain-Morrey序列空间的结构性质、与经典ℓ^p空间的关系及对偶性，揭示了其可分性、嵌入关系及对偶空间的构造。

摘要：

我们研究了最近作为Morrey型空间离散 counterpart 引入的离散Bourgain-Morrey序列空间ℓ^{p}_{q,r}(ℤ)。我们证明c_{00}在ℓ^{p}_{q,r}中稠密，因此这些空间是可分的。我们建立了嵌入ℓ^{1}↪ℓ^{p}_{q,r}↪ℓ^{r}对于r>1，而当r=1时有ℓ^{p}_{q,1}=ℓ^{1}。对于每个p，恒等式ℓ^{p}_{q,p}=ℓ^{p}导致在ℓ^{p}上存在不可数多个等价范数。我们还引入了一个块空间作为ℓ^{p}_{q,r}的自然预对偶空间，并证明对偶性(ℓ^{p}_{q,r})^{*}=h^{p'}_{q',r'}，由此可得对于1<p<q<∞和1<r<∞时的 reflexivity。这项工作通过完全刻画其结构和对偶性，完成了离散Bourgain-Morrey理论的基础阶段。

英文摘要：

We study the discrete Bourgain-Morrey sequence spaces $\ell^{p}_{q,r}(\mathbb{Z})$, recently introduced as discrete counterparts of Morrey-type spaces. We show that $c_{00}$ is dense in $\ell^{p}_{q,r}$, hence the spaces are separable. We establish embeddings $\ell^{1}\hookrightarrow \ell^{p}_{q,r}\hookrightarrow \ell^{r}$ for $r>1$, while for $r=1$ one has $\ell^{p}_{q,1}=\ell^{1}$. For each $p$, the identity $\ell^{p}_{q,p}=\ell^{p}$ yields uncountably many equivalent norms on $\ell^{p}$. We also introduce a block space as a natural predual of $\ell^{p}_{q,r}$ and prove the duality $(\ell^{p}_{q,r})^{*}=\mathrm{h}^{p'}_{q',r'}$, from which reflexivity follows for $1<p<q<\infty$ and $1<r<\infty$. This work completes the foundational stage of the discrete Bourgain-Morrey theory by fully characterizing its structure and duality.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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