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If $u \in \mathcal D'(\mathbb R^n)$, $u$ is homogeneous of degree $0$ and rotational invariant, it is necessarily that $u$ is a constant? (Since if $u \in C^\infty$, the conclusion obviously hold.)

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Could you state your definition of homogeneous distribution? I assume it is rolled down to the test function, but I would ask for your specification. –  Vobo Apr 23 '12 at 20:22

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Any homogeneous distribution u of degree a in $\mathbb{R}^{n}$ can be writen as the form $$u=r^{a}f$$ where $r=|x|$,and $f\in\mathcal D'(S^{n-1})$.So,in your case, distributions on the unit sphere which is symmetric will satisfy. see aslo http://archive.numdam.org/article/ASNSP_1972_3_26_1_117_0.pdf

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