## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Spherical associated homogeneous distributions on $R^{n}$

Ghislain R. Franssens

#### 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 $R^{n}$.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 17, Number 5 (2010), 781-806.

Dates
First available in Project Euclid: 14 December 2010

https://projecteuclid.org/euclid.bbms/1292334055

Digital Object Identifier
doi:10.36045/bbms/1292334055

Mathematical Reviews number (MathSciNet)
MR2777770

Zentralblatt MATH identifier
1220.46027

#### Citation

Franssens, Ghislain R. Spherical associated homogeneous distributions on $R^{n}$. Bull. Belg. Math. Soc. Simon Stevin 17 (2010), no. 5, 781--806. doi:10.36045/bbms/1292334055. https://projecteuclid.org/euclid.bbms/1292334055