## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

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

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.36045/bbms/1292334055

**Mathematical Reviews number (MathSciNet)**

MR2777770

**Zentralblatt MATH identifier**

1220.46027

**Subjects**

Primary: 46F05: Topological linear spaces of test functions, distributions and ultradistributions [See also 46E10, 46E35] 46F10: Operations with distributions 31B99: None of the above, but in this section

**Keywords**

Spherical associated homogeneous distribution Pullback Potential theory

#### 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