Bourgain-Morrey sequence spaces: Structural properties, relations to classical $\ell^{p}$ spaces and duality
Francisco Alejandro Villegas Acuña
TL;DR
This work develops the full structural theory of the discrete Bourgain-Morrey sequence spaces $\ell^{p}_{q,r}(\mathbb{Z})$, establishing that $c_{00}$ is dense and that the spaces sit between $\ell^{1}$ and $\ell^{r}$ with $\ell^{p}_{q,1}=\ell^{1}$ and $\ell^{p}_{q,p}=\ell^{p}$. It introduces a natural block predual $\mathrm{h}^{p'}_{q',r'}(\mathbb{Z})$ and proves the duality $(\ell^{p}_{q,r})^{*}=\mathrm{h}^{p'}_{q',r'}$, yielding reflexivity in the interior range $1<p<q<\infty$, $1<r<\infty$. The theory also produces uncountably many equivalent norms on classical $\ell^{p}$ spaces via the $\ell^{p}_{q,r}$ construction and clarifies the endpoint behavior at $r=\infty$ by connecting to discrete Morrey spaces. Applications to operators and linear/nonlinear difference equations demonstrate the practical utility of the framework for discrete harmonic analysis and operator theory.
Abstract
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.