We're supposed to show that for $\lambda\geq d-1$, there does not exist a weak derivative $v\in L^1_{loc}(\mathbb{R}^d)$ of $|x|^{-\lambda}$. Im really stuck here. Here's what I came up with so far: Let $\phi\in C_c^\infty$ and $R>0$ large enough, such that $\mathrm{supp}(\phi)\subset B(0,R)$. Now let $1\leq j\leq d$. I want to show that for $\lambda\geq d-1$, $|x|^{-\lambda}\partial_j\phi$ is not integrable and I figured that if it fails to be so, it's going to be at the origin. Hence by choosing $R$ as above I don't have to worry about the terms outside of a large enough ball. So$$\int_{\mathbb{R}^d}|x|^{-\lambda}\partial_j\phi=\int_{|x|<R}|x|^{-\lambda}\partial_j\phi=\int_0^Rr^{d-\lambda-1}\left(\int_{S^{d-1}}\partial_j\phi(ry)\ d\sigma(y)\right)\ dr,$$where I used spherical coordinates in the last equality and $\sigma$ is the surface measure on $S^{d-1}$. Now I'd like to argue that this integral diverges because $r^{d-\lambda-1}$ is not integrable on $[0,R]$ (at least when $\lambda>d-1$). But here I don't know how to proceed. Maybe someone can give me a hint or point out a different approach that I can try: Any help is appreciated, thanks.
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