Bourgain-Morrey sequence spaces: structural properties, relations to classical $\ell^{p}$ spaces and duality
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910012677292032 |
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| author | Acuña, Francisco Alejandro Villegas |
| author_facet | Acuña, Francisco Alejandro Villegas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_00322 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bourgain-Morrey sequence spaces: structural properties, relations to classical $\ell^{p}$ spaces and duality Acuña, Francisco Alejandro Villegas Functional Analysis 46B45, 46B10, 46B15, 46B70, 46E30, 42B35 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. |
| title | Bourgain-Morrey sequence spaces: structural properties, relations to classical $\ell^{p}$ spaces and duality |
| topic | Functional Analysis 46B45, 46B10, 46B15, 46B70, 46E30, 42B35 |
| url | https://arxiv.org/abs/2602.00322 |