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The scaling property of the Fourier transform extends by duality toConsequently a homogeneous distribution of degree $\sigma$ transforms into one of degree $-n-\sigma$.

The scaling property of the Fourier transform gives, first for Schwartz functions and then by duality,If the homogeneous distribution $u$ has degree $\sigma$, thenPutting $s=1/t$ yields ${\delta }_s\hat{u}=s^{-n-\sigma }\hat{u}$. Thus the Fourier transform of a homogeneous distribution has degree $\boxed{-n-\sigma }$.

Because $0<\alpha <n$, $|x{|}^{-\alpha }$ is locally integrable at the origin and has only polynomial growth at infinity, so it defines a tempered distribution. It is a homogeneous distribution of degree $-\alpha$ and is radial. Its Fourier transform is therefore radial and homogeneous of degree $\alpha -n$, so it must have the form $c_{\alpha }|\lambda {|}^{\alpha -n}$. In particular, $\boxed{\beta =n-\alpha }$.

To determine the constant, use the stated Gamma integral representation, Fubini's theorem, and the Fourier transform of a Gaussian:The change of variables formula $s=|\lambda {|}^2/(4\tau )$ then givesThis is precisely the Fourier transform of the Riesz kernel.