negative Sobolev space contains $L^1$ for a compact domain

Question

I'd like to use something like Aubin-Lions lemma for the following spaces: $$ C^{0, \alpha}(B) \subset L^1(B) \subset W^{-1, q}(B),$$ with $B \subset \mathbb{R}^n$ being a compact, say a closed ball and $q$ is some constant s.t. $q < 3/2$ but I guess we will need $q \leq 1$. The first inclusion is trivial, since all continuous functions are integrable. I also need it to be compact, but that's ok too, since we know that $C^{0,\alpha} \subset C^{0,\beta}$ is compact for $\beta < \alpha$ so I can just use that I guess - would appreciate a correction if this doesn't work. Nevertheless my main problem is the second injection. Negative Sobolev means that my function $f \in L^1$ should be in $L^q$ itself so I guess $q$ should be smaller/equal to $1$. then again $f$ also has to be a weak derivative of something from $L^q$. If $q < 1$ then since my domain is bounded then $f \in L^q$ so there's no problem with that part, the only issue is finding a function $g \in L^q$ s.t. $$\sum_i^n a_i \partial_{x_i} g = f$$ can anyone provide some insight? does this even hold? if so, what are the necessary/sufficient constraints on $q$? a reference would be perfectly fine answer

Answer 1

The negative-order Sobolev spaces are usually defined as dual spaces: $W^{-1,q}=(W_0^{1,p})^*$ where $1/p+1/q=1$. This works only for $q>1$. Another definition, which makes sense also for $q=1$, is that $W^{-1,q}=\{\operatorname{div}\vec F: \vec F\in L^q\}$ where $\vec F$ is a vector field. The derivatives forming the divergence of $\vec F$ are understood in the sense of distributions. The norm of a distribution $T\in W^{-1,q}$ is the infimum of $\|\vec F\|_q$ over all $\vec F$ such that $T=\operatorname{div}\vec F$. See Sobolev spaces by Adams. Note that neither definition makes sense for $q<1$.

However, you don't need $q<1$. Nate Eldredge gave a hint in the right direction: the embedding $L^1\subset W^{-1,q}$ holds for $q<\frac{n}{n-1}$. This statement simply means that we can integrate $L^1$ functions against functions in $W^{1,q'}$. Which is true because $W^{1,q'}\subset L^\infty$ by Morrey-Sobolev embedding.