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

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.

💡 Research Summary

This paper provides a comprehensive functional‑analytic study of the discrete Bourgain‑Morrey sequence spaces ℓ⁽ᵖ⁾{q,r}(ℤ), which were recently introduced as the discrete analogues of Bourgain‑Morrey function spaces. The authors begin by recalling the three equivalent norms that define ℓ⁽ᵖ⁾{q,r}: the centered‑interval norm, the dyadic‑interval norm, and the dyadic‑length‑interval norm. They prove that these norms are mutually comparable, guaranteeing that the space is well defined regardless of the chosen formulation.

The first major result establishes that the space of finitely supported sequences c₀₀ is dense in ℓ⁽ᵖ⁾{q,r}. By selecting a finite collection of dyadic intervals that capture most of the norm and truncating a given sequence to this finite set, the authors construct a c₀₀ approximation arbitrarily close in the ℓ⁽ᵖ⁾{q,r} norm. Consequently ℓ⁽ᵖ⁾_{q,r} is separable, a property essential for many subsequent arguments.

Next, the paper investigates continuous embeddings. Using the unit vector e₀ and the convolution invariance of ℓ⁽ᵖ⁾{q,r}, they show that every ℓ¹‑sequence belongs to ℓ⁽ᵖ⁾{q,r} with a norm estimate ‖y‖{p,q,r} ≤ ‖e₀‖{p,q,r}‖y‖₁. Conversely, by examining the dyadic interval of length one (I(0,k) = {k}), they prove that any element of ℓ⁽ᵖ⁾{q,r} lies in ℓʳ and satisfies ‖x‖r ≤ ‖x‖{p,q,r}. Thus for r>1 we have the strict chain ℓ¹ ⊂ ℓ⁽ᵖ⁾{q,r} ⊂ ℓʳ, while for r=1 the space collapses to ℓ¹ (i.e., ℓ⁽ᵖ⁾_{q,1}=ℓ¹) with equivalent norms.

A particularly striking observation is the case r=p. The authors prove that ℓ⁽ᵖ⁾_{q,p}=ℓᵖ for every q>p, which yields uncountably many equivalent norms on the classical ℓᵖ space. Each choice of q>p produces a distinct norm that is nevertheless equivalent to the standard ℓᵖ norm, offering a rich family of renormings that may have geometric or interpolation implications.

The core of the paper is the duality theory. The authors introduce a block space h^{p’}{q’,r’}(ℤ) as a natural predual, where p’, q’, r’ are the Hölder conjugates of p, q, r respectively. Elements of h^{p’}{q’,r’} are sequences whose dyadic block ℓ^{p’}‑norms, weighted by the factor |I(j,k)|^{(1/q)-(1/p)}, belong to ℓ^{r’}. By constructing a bilinear pairing between ℓ⁽ᵖ⁾{q,r} and h^{p’}{q’,r’} and applying the Hahn‑Banach theorem, they establish an isometric isomorphism (ℓ⁽ᵖ⁾{q,r})* ≅ h^{p’}{q’,r’}. The converse identification (h^{p’}{q’,r’})* ≅ ℓ⁽ᵖ⁾{q,r} follows similarly. This duality holds for the interior range 1<p<q<∞ and 1<r<∞, and immediately yields reflexivity of ℓ⁽ᵖ⁾_{q,r} in that regime.

The paper also discusses limiting cases. When r=1 the space reduces to ℓ¹, whose dual is ℓ^∞, consistent with the classical theory. The case r=∞ remains open; the authors note that a full description of the dual space in this extreme is a natural direction for future work.

In the final sections, the authors summarize the embedding picture, emphasizing that ℓ⁽ᵖ⁾_{q,r} serves as an intermediate space between ℓ¹ and ℓʳ, and that the block predual provides a concrete functional‑analytic framework mirroring the continuous Bourgain‑Morrey theory. They also hint at potential applications to operator theory, interpolation, and discrete difference equations, suggesting that the structural results obtained here lay a solid foundation for such developments.

Overall, the paper successfully completes the foundational stage of discrete Bourgain‑Morrey theory by delivering a full characterization of separability, embedding relations, renorming phenomena, and a precise duality formula, thereby enriching the landscape of Banach sequence spaces.