Homogeneous generalized functions which are rotation invariant

Abstract

We characterize generalized functions including distributions and ultradistributions which are rotation invariant and homogeneous as follows:

If $u$ is a generalized function in $\mathbf{R}^{n}$ with $n \geq 2$ which is rotation invariantand homogeneous of real degree $k$ then it can be written as \[ \begin{array}{ccc} u&=&\left\{ \begin{array}{cl}a|x|^{k}+b\Delta ^{\frac{-n-k}{2}}\delta , & \text{if } -n-k \text{ is an even nonnegative integer,}\\ a|x|^{k}, & \text{otherwise.}\end{array}\right. \end{array} \] In addition, we find a structure theorem of rotation invariant ultradistributions with support at the origin.