A Bourgain-Gromov problem on non-compact Sobolev-Lorentz embeddings

TL;DR

This work addresses the non-compact Sobolev embeddings into optimal Lorentz spaces, specifically the mapping $W_0^m L^{p,q}(\,\Omega\,)\hookrightarrow L^{p^*,r}(\,\Omega\,)$ with $p^*=rac{dp}{d-mp}$ and $0<q<r\le\infty$. The authors develop a Besov-space and sequence-space reduction, using wavelet decompositions to connect the Sobolev–Lorentz embedding to corresponding Besov embeddings and then to $\ell^q(\ell^{p})$–type sequence spaces, enabling explicit Bernstein-number bounds. They prove the embedding is finitely strictly singular and provide upper bounds on Bernstein numbers, deriving sharp asymptotics in the regime $p<q<r<p^*$ and showing the unbounded-domain setting and Lorentz targets can be treated beyond previous results. The new approach yields a simpler, more general framework than prior work by Bourgain–Gromov and Lang–Mihula, clarifying how non-compactness is quantified via Bernstein numbers rather than the ball measure of non-compactness, with implications for the fine structure of Sobolev embeddings into Lorentz scales.

Abstract

We study the non-compact Sobolev embeddings into the optimal scale of Lorentz spaces, $W_0^mL^{p,q}(Ω) \to L^{\frac{dp}{d - mp},r}(Ω)$, where $Ω\subseteq \mathbb{R}^d$, $1 \le m \le d$ and $0<q<r\le\infty$ with $1<p<\frac dm$ or $p=q=1$. We show that these embeddings are finitely strictly singular with certain upper bounds on the decay rate of the Bernstein numbers. We reduce the Sobolev embeddings to embeddings of Besov spaces and sequence spaces, which simplifies the previous methods by Bourgain-Gromov and Lang-Mihula.

Theorems & Definitions (10)

• Theorem 1
• Lemma 2
• proof
• Remark 3
• Remark 4
• proof : Proof of Theorem \ref{['Thm::MainThm']}
• Proposition 5
• Lemma 6
• proof
• proof : Proof of Proposition \ref{['Prop::SobSharp']}