# Weak-star approximation of smooth functions in weak $L^p$-space

It is well known that the weak space $$L^{p,\infty}$$ has less density property contrary to standard $$L^p$$ space. Related to this one, I'm struggling to prove the following statement which is given in the paper of Baker-Seregin-Sverak:

Proposition. Let $$u_0 \in L^{3,\infty}$$ be divergence-free in the sense of distributions. Then there exists a sequence $$u_0^{(k)} \in C_{0,0}^\infty(\mathbb{R}^3)$$ such that $$u_0^{(k)}\rightarrow u_0\quad \text{weakly star in } L^{3,\infty}.$$

Here $$C_{0,0}^\infty(\mathbb{R}^3)$$ is the space of all smooth vector fields with compact support whose divergence is free.

I have no idea to prove the above statement. Approximation by smooth functions is easy by using mollification, but I have no idea to obtain a suitable sequence as stated in the proposition. In the case of $$L^p$$ with $$1, by using the Hahn-Banach theorem and De Rham's theorem, we can show that $$L^p_\sigma = \left\{ u \in L^p : \int_{\mathbb{R}^3} u\cdot \nabla \phi \,dx=0\quad \text{for all } \phi \in D^{1,p'}\right\}.$$ Here $$L^p_\sigma$$ is the closure of $$C_{0,0}^\infty$$ under $$L^p$$-norm and $$D^{1,p'}$$ is the space of all functions $$u$$ such that $$\nabla u \in L^{p'}$$.

The statement as written above is incorrect. It should be weak-$$\ast$$ convergence instead of strong convergence. This is one of the key features of $$L^{p,\infty}$$: Test functions are not dense. As a general comment, in the Navier-Stokes context, local well-posedness holds for initial data in the strong closure of divergence-free test functions in $$L^{3,\infty}$$ but probably not for general large divergence-free fields in $$L^{3,\infty}$$. For example, non-uniqueness of large self-similar solutions is conjectured in the papers of Jia-Sverak, with numerical evidence later given by Guillod-Sverak.
To do the weak-$$\ast$$ approximation, my suggestion would be to mollify, apply a smooth cut-off, and then use Bogovskii's operator to add a correction which makes it divergence free (see Chapter 3 of Galdi's book for an extensive discussion, and there is also a statement in Tsai's book). You can use real interpolation to bound the Bogovskii operator on Lorentz spaces. This procedure is very useful for localizing divergence-free functions.