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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.

flag	asked Apr 1 2011 at 19:11 John H 107●3

Michael Renardy · Accepted Answer · 2011-04-01 19:54:55Z UTC

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

answered Apr 1 2011 at 19:54

Michael Renardy
4,771●1●10●17

	Where could I learn why this is the solution? – John H Apr 2 2011 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 2011 at 19:50

Division of distributions by polynomials.

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