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Can you tell me for every $\alpha \in \mathbb{R}$, whether there is a non-zero homogeneous and rotation-invariant distribution on $\mathbb{R}^n$ with degree of homogeneity $\alpha$?

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Yes. When $\alpha>-n$, the obvious candidate $|x|^\alpha$ works. Also, if $f$ is a distribution that is homogeneous of degree $\alpha$, then $\Delta f$ is a homogeneous distribution of degree $\alpha-2$ (and rotational invariance is preserved). This allows you to cover the range $\alpha\le -n$ as well.

There is one exception: we do not get a nonzero distribution with $\alpha=-2$ by taking the Laplacian of $|x|^0\equiv 1$. Take $\Delta \log |x|$ instead.

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