Abstract
A structure theorem for spherically symmetric associated homogeneous distributions (SAHDs) based on $R^{n}$ is given. It is shown that any SAHD is the pullback, along the function $\left\vert \mathbf{x}\right\vert ^{\lambda }$,\ $\lambda \in \mathbf{C}$, of an associated homogeneous distribution (AHD) on $R$. The pullback operator is found not to be injective and its kernel is derived (for $\lambda =1$). Special attention is given to the basis SAHDs, $D_{z}^{m}\left\vert \mathbf{x}\right\vert ^{z}$, which become singular when their degree of homogeneity $z=-n-2p$, $\forall p\in \mathbb{N}$. It is shown that $\left( D_{z}^{m}\left\vert \mathbf{x} \right\vert ^{z}\right) _{z=-n-2p}$ are partial distributions which can be non-uniquely extended to distributions $\left( \left( D_{z}^{m}\left\vert \mathbf{x}\right\vert ^{z}\right) _{e}\right) _{z=-n-2p}$ and explicit expressions for their evaluation are derived. These results serve to rigorously justify distributional potential theory in $R^{n}$.
Citation
Ghislain R. Franssens. "Spherical associated homogeneous distributions on $R^{n}$." Bull. Belg. Math. Soc. Simon Stevin 17 (5) 781 - 806, december 2010. https://doi.org/10.36045/bbms/1292334055
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