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

We study the non-compact Sobolev embeddings into the optimal scale of Lorentz spaces, $W_0^m{L}^{p,q}(\mathrm{\Omega })\to L^{\frac{dp}{d-mp},r}(\mathrm{\Omega })$, where $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$, $1\le m\le d$ and $0<q<r\le \mathrm{\infty }$ with $1<p<\frac{d}{m}$ 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.