# Division of distributions by polynomials.

I'm trying to solve the equation

$(1-|x|^2)T = 0$, where $T$ is a tempered distribution. I know how to do this (it is a common exercise) in dimension $1$. How can I solve it in higher dimensions?

Thank you very much.

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## 1 Answer

The solution is the direct product of $\delta(|x|-1)$ and any distribution on the sphere.

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Where could I learn why this is the solution? – John H Apr 2 '11 at 0:01
First prove that T has compact support on the sphere. Any test function localized near the sphere can be split into a part that vanishes on the sphere and a part that is radially constant near the sphere. Now consider how T acts on each part. – Michael Renardy Apr 5 '11 at 19:50