The fourier transform of homogeneous distribution and related topics

Question

When we have a distribuion $u\in \mathcal{D}'(R^n)$,and the restriction to $R^{n}\backslash{0}$ is homogeneous of degree a,we have $u \in \varphi'$ and $\widehat u$ is of degree(-n-a) in $R^{n}\backslash{0}$ .So it's easy to get when $-n< a <0$, $\widehat |x|^a=c_{a,n}|x|^{-n-a}$. my question is what's the result when $a>0$ ?Indeed, it's homogeneous of degree -n-a in$R^n$ and when a=n=1,we have $\widehat|x|=C p.v.|x|^{-2}$,where C is some constant. Futhermore,when we consider the wavefront set,it has the following:if u is a homogeneous distribuion in $R^{n}\backslash{0}$ ,then $(x,\xi) \in WF(u)\Leftrightarrow (\xi,-x) \in WF(\widehat u)$,where $x \neq 0$ , $\xi \neq 0$. Are there other interesting properties related to homogeneous distribuion ?

Answer 1

In the first place your presentation must be clarified. An homogeneous distribution $u$ of degree $a$ on $\mathbb R^n$ is characterized by $$\forall \lambda >0,\quad u(\lambda x)=\lambda^au(x),\qquad \text{i.e}\quad\langle u(x),\phi(x/\lambda\rangle\lambda^{-n}=\lambda^{a} \langle u(x),\phi(x\rangle, $$ which is also equivalent to the so-called Euler equation $x\cdot\partial_xu=au$. It is not difficult (but not trivial) to prove that an homogeneous distribution on $\mathbb R^n$ is a temperate distribution. Simple examples include in one dimension $pv(1/x)$ (degree $-1$), $H(x)$ (degree $0$), $(x+i0)^a$ (degree $a$), and in $n$ dimensions $\delta_0$ (degree $-n$), $\vert x\vert^a$ for $\Re a>-n$ (degree $a$),$\partial_x^\alpha \vert x\vert^a$ (degree $a-\vert \alpha\vert$). For $a\in \mathbb C$, $$ \frac{x_+^a}{\Gamma(a+1)}\text{is an homogeneous distribution with degree $a$, entire with respect to $a$.} $$

On the other hand, you may consider homogeneous distributions on any cone of $\mathbb R^n$, e.g. on $\mathbb R^n\backslash 0$ and wonder if you can extend these distributions to homogeneous distributions on the whole $\mathbb R^n$. In particular the following result holds: if $u$ is an homogeneous distribution on $\mathbb R^n\backslash 0$ with degree $a$ which is not an integer $\le -n$, then there is a unique extension of $u$ to an homogeneous distribution on the whole $\mathbb R^n$.

Last but not least, if $u$ is an homogeneous distribution of degree $a$ on $\mathbb R^n$, it is temperate and its Fourier transform is indeed homogeneous on $\mathbb R^n$ with degree $-a-n$. In particular, you may consider $\vert x\vert ^a$ which is a radial distribution homogeneous of degree $a$ on $\mathbb R^n\backslash 0$; for $a$ avoiding integer values, it has a unique extension as described above as an homogeneous distribution with degree $a$ on $\mathbb R^n$ and its Fourier transform is also radial and homogeneous with degree $-a-n$, that is coincides with $ c\vert \xi\vert^{-a-n} $ on $\mathbb R^n\backslash 0$.