Q:Suppose that $u$ is a $\mathcal{C}^{\infty}$ function on $\mathbb{R}^{n} \backslash\{0\}$ that is homogeneous of degree $-n+i \tau, \tau \in \mathbb{R} .$ Prove that the operator given by convolution with $u$ maps $L^{2}\left(\mathbb{R}^{n}\right)$ to $L^{2}\left(\mathbb{R}^{n}\right)$.
A distribution in $\mathscr{S}^{\prime}\left(\mathbb{R}^{n}\right)$ is called homogeneous of degree $\gamma \in \mathbb{C}$ if for all $\lambda>0$ and for all $\varphi \in \mathscr{S}\left(\mathbb{R}^{n}\right),$ we have $$\left\langle u, \delta^{\lambda} \varphi\right\rangle=\lambda^{-n-\gamma}\langle u, \varphi\rangle,$$ where $\delta$ is dilation.
It seems I need to prove $u*f\in L^{2}\left(\mathbb{R}^{n}\right)$ for all $f\in L^{2}\left(\mathbb{R}^{n}\right)$, I am stuck since I have no idea how to combine translation and dilation. Sincerely thanks for your help!