From Richard Melrose, From microlocal to global analysis, Chapter 1, Problem 11.
Suppose $u\in L^{2}(\mathbb{R}^{n})$ and that $$ D_{1}D_{2}\cdots D_{n}\mu\in (1+|x|)^{-n-1}L^{2}(\mathbb{R}^{n}) $$ where the derivatives are defined using Problem 10 (Melrose meant weak derivative). Using repeated integration, show that $u$ is necessarily a bounded continuous function. Conclude further that for $u\in S'(\mathbb{R}^{n})$, if we have $$ D^{\alpha}\mu\in (1+|x|)^{-n-1}L^{2}(\mathbb{R}^{n}) \quad \forall |\alpha|\le k+n, $$ then $D^{\alpha}\mu$ is bounded and continuous for $|\alpha|\le k$. (This is a weak form of the Sobolev embedding theorem.)
I do not really know what is the best way to approach this problem. My efforts at $n=1$ only let me get $\int x^{2}\mu^{2}<\infty$. By Fourier transform we have $F(x\mu)=\frac{\partial}{\partial \xi}F(u)$, and effectively we have $Fu, D(Fu)\in L^{2}$. But this is not sufficient to show that $u$ is bounded and continuous. To do that I need to show that $Fu$ is bounded. So I am looking for a hint as I got stuck for quite a few hours.
I know the proof of the classical Sobolev embedding theorem as well as a proof of the weak Sobolev embedding on the Fourier side. But I have never seen it in this form before. My problem is I do not know how to treat both $u$ without Fourier transform and $(1+|x|)^{2}$ at the same time.