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user23078
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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 WF(\widehat u)$$(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 ?

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 $(x,\xi) \in WF(u)\Leftrightarrow WF(\widehat u)$,where $x \neq 0$ , $\xi \neq 0$. Are there other interesting properties related to homogeneous distribuion ?

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 ?

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user23078
user23078
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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$-n< a <0$, $\widehat |x|^a=c_{a,n}|x|^{-n-a}$. 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$$R^{n}\backslash{0}$ ,then $(x,\xi) \in WF(u)\leftrightarrow WF(\widehat u)$$(x,\xi) \in WF(u)\Leftrightarrow WF(\widehat u)$,where $x \neq 0$ , $\xi \neq 0$. Are there other interesting properties related to homogeneous distribuion ?

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 $R^n$ ,then $(x,\xi) \in WF(u)\leftrightarrow WF(\widehat u)$ Are there other interesting properties related to homogeneous distribuion ?

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 $(x,\xi) \in WF(u)\Leftrightarrow WF(\widehat u)$,where $x \neq 0$ , $\xi \neq 0$. Are there other interesting properties related to homogeneous distribuion ?

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user23078
user23078
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The fourier transform of homogeneous distribution and related topics

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 $R^n$ ,then $(x,\xi) \in WF(u)\leftrightarrow WF(\widehat u)$ Are there other interesting properties related to homogeneous distribuion ?