Fourier transform of $1/|x|^{\alpha}$.

Denoting by $\widehat{\cdot}$ the Fourier transform in $\mathbb R^d$ and letting $$ f_\lambda(x)\equiv f(\lambda x), $$ then, by definition of the Fourier transform, \begin{align} \widehat {f_\lambda}(k)&=\int_{\mathbb R^n}e^{-ik\cdot x}f(\lambda x)dx\\ &=\int_{\mathbb R^n}e^{-ik\cdot x/\lambda}f(x)\lambda^{-d}dx\\ &=\lambda^{-d}\widehat f(k/\lambda). \end{align} Let $f(x)=F(|x|)$ be a radial function; if $M\in SO(d)$, then $f_M(x)\equiv f(Mx)=f(x)$ and hence, since $\widehat{ f_M}(k)=\widehat f(Mk)$, we have $\widehat {f_M}= \widehat f$, which in turn implies that also $\widehat f$ is radial. Now, suppose $f(x)=|x|^{-\alpha}$, for $0<\alpha<d$, then $$ f_\lambda(x) = \lambda^{-\alpha}f(x) $$ and, by the above remark, $$ \widehat f (k/\lambda)= \lambda^{d}\widehat {f_\lambda}(k) = \lambda^{d-\alpha}\hat f(k). $$ The only radial $\widehat f$ homogeneous of degree $-d+\alpha$ is $$ \widehat f (k) = |k|^{\alpha-d}, $$ up to constants.

Note that $f$ can be split as the sum of two pieces: let $B_a$ be the ball of radius $a$ centred at the origin, then \begin{align} f&\equiv u+v\\ &= \chi_{B_a}(x)\frac{1}{|x|^{\alpha}}+\left(1-\chi_{B_a}(x)\right)\frac{1}{|x|^{\alpha}}; \end{align} $u$ lies in $L^1(\mathbb R^d)$, for $0<\alpha<d$, and $v$ lies in $L^2(\mathbb R^d)$, for $\alpha>d/2$. This ensures $\widehat u \in L^\infty(\mathbb R^d)$ and $\widehat v\in L^2(\mathbb R^d)$ and that $\widehat f$ needs indeed to be a function in $L^1_{\text{loc}}(\mathbb R^d)$ from an abstract point of view. So $f,\widehat f\in L^1_{\text{loc}}(\mathbb R^d)$.

More generally, $f$ lies in $\mathscr S'(\mathbb R^d)$, which is stable under Fourier transform.

Following Brightsun solution we have $\widehat f (\xi) = k/|x|^{n-\alpha}$. Let $g(x) = e^{-\pi x^2}$, we have $\widehat g = g$. Since $$ \int_{\mathbb R^n} \widehat{f}(x)g(x)dx = \int_{\mathbb R^n} f(x)\widehat{g}(x)dx,$$ we have $$ \int_{\mathbb R^n} \frac{k}{|x|^{n-\alpha}}e^{-\pi x^2}dx = \int_{\mathbb R^n} \frac{1}{|x|^\alpha}e^{-\pi x^2}dx.$$ In polar coordinates we have: $$\int_{\mathbb R^n} \frac{1}{|x|^{\alpha}}e^{-\pi x^2}dx = \sigma(\mathbb S^{n-1})\int_{0}^{\infty} \frac{e^{-\pi r^2}r^{n-1}}{r^{\alpha}} = \sigma(S^{n-1})\int_{0}^{\infty} e^{-\pi r^2}r^{n-1-\alpha} dr,$$ where $\sigma$ is the measure on the sphere. Taking $u = \pi r^2$ we obtain $$\sigma(S^{n-1})\int_{0}^{\infty} e^{-\pi r^2}r^{n-1-\alpha} dr = \frac{\sigma(S^{n-1})}{\pi^{\alpha/2}}\int_{0}^{\infty}u^{\alpha/2 -1} e^{-u}du = \frac{\sigma(S^{n-1})\Gamma(\alpha/2)}{\pi^{\alpha/2}}.$$ Analogously, we can deduce $$\int_{\mathbb R^n} \frac{k}{|x|^{n-\alpha}}e^{-\pi x^2}dx = \frac{k\sigma(S^{n-1})\Gamma((n-\alpha)/2)}{\pi^{(n-\alpha)/2}}$$ and we have $$k=\frac{\Gamma(\alpha/2)\pi^{(n-\alpha)/2}}{\Gamma((n-\alpha)/2)\pi^{\alpha/2}}.$$ Therefore, the identity $$C_{\alpha}\int_{\mathbb R^n} \frac{1}{|x|^\alpha} \phi(x) dx = C_{n-\alpha}\int_{\mathbb R^n} \frac{1}{|x|^{n-\alpha}} \widehat{\phi}(x) dx$$ holds.