= Spherical surface measure convolution
{title2=$u_a*u_b$}
For the geometric <surface delta distribution> on the radius-$a$ sphere in $\mathbb R^3$, $\langle u_a,\phi\rangle=a^2\int_{S^2}\phi(a\omega)\,d\sigma(\omega)$. Angular integration gives the <Fourier transform> $\widehat u_a(\xi)=4\pi a\sin(a|\xi|)/|\xi|$, with removable value $4\pi a^2$ at zero. The <convolution of distributions with a compactly supported factor> gives the <regular distribution>
$$
u_a*u_b=\frac{2\pi ab}{|x|}\mathbf1_{\{|a-b|<|x|<a+b\}}.
$$
This density has total mass $16\pi^2a^2b^2$. Values on endpoint spheres do not affect the <distribution>. When $a=b$, the inverse-distance singularity is locally integrable in three dimensions and is not an additional point mass. The annulus expresses the triangle inequality for the sum of two vectors of fixed lengths.
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