There is a definition in distribution theory I came across. If $f$ is a distribution on $R^n$, then it is homogeneous of degree $K$ if $f(rx)=r^Kf(x)$ for all $x$ and $r>0$.
I am trying to show that $\delta$ has degree $-n$. But I am having difficulties as $\delta(x)$ and $\delta(rx)$ always coincide.
Furthermore there is a distributional derivative given by $(I_{(x>0)}\log(x))'$. It is hard for me to see why this is not homogeneous. Furthermore this function supposedly agrees with a function a.e that is homogeneous of degree -1. What is this function? can it be found this explicitly?
For $f=1_{ x >0}\log x\in D'(\Bbb{R})$ and $r>0$ $$f(rx)=f+1_{x >0}\log r$$ $$f'(rx)= r^{-1} (f(rx))'=r^{-1} f' + \color{red}{r^{-1} \delta(x)\log r}$$