Let $u_k$ be a sequence of functions in $C^{\infty}_c(B_1),$ where $B_1 \subset \mathbb{R}^n$ is the unit ball.
Then, if $$\sup_{k} \|u_k\|_{L^1(B_1)} < \infty \quad \text{and} \quad \sup_{k} \|\nabla u_k\|_{L^{1,\infty}(B_1)}<\infty,$$ is it true that there is a subsequence converging strongly in some function space?
In the above, the weak-$L^1$ norm of a function $v$ is defined as $$\|v\|_{L^{1,\infty}(B_1)} = \sup_{t > 0} t \, |\{x \in B_1 \,\colon \, |\nabla v(x)| > t\}|.$$
More concretely, is it true that, up to a subsequence, we have $$u_k \to u \quad \text{(say) in } L^{1}(B_1)?$$
For instance, this particular result would be true if we had an inequality of the form $$[u_k]_{W^{s,1}(B_1)} \leq C \|\nabla u_k\|_{L^{1,\infty}(B_1)}$$ for some $s \in (0,1),$ where $$[v]_{W^{s,1}(B_1)} = \int_{B_1} \int_{B_1} \dfrac{|v(x)-v(y)|}{|x-y|^{n+s}} d x d y$$ is the Gagliardo seminorm. Indeed, a known compactness result for fractional Sobolev spaces states that $$\sup_{k} \|u_k\|_{L^1(B_1)} + [u_k]_{W^{s, 1}(B_1)} < \infty \quad \Rightarrow \quad u_k \to u \text{ in } L^1(B_1) \text{ up to a subsequence.}$$
Is such an inequality available?
