A question about homogeneous distribution

Question

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 $\delta^{\lambda} \varphi(x)=\varphi(\lambda x)$. Now suppose that $u \in C^{\infty}\left(\mathbb{R}^{n} \backslash\{0\}\right)$ is homogeneous of degree $-n+i \tau, \tau \in \mathbb{R} .$ How to prove that the operator given by convolution with $u$ maps $L^{2}\left(\mathbb{R}^{n}\right)$ to $L^{2}\left(\mathbb{R}^{n}\right)$.

Answer 1

Let $u$ be a smooth function on $\mathbb R^n\backslash\{0\}$ homogeneous with degree $\lambda$ (on $\mathbb R^n\backslash\{0\}$). If $\lambda$ is not an integer $\le -n$, then $u$ can be uniquely extended to a tempered distribution homogeneous with degree $\lambda$. Moreover, the Fourier transform of an homogeneous distribution with degree $\lambda$ is an homogeneous distribution of degree $-\lambda-n$.

As a result, the Fourier transform of your $u$ is homogeneous with degree $n-i\tau-n=-i\tau$ when $\tau\in \mathbb R^*$, so is in one dimension a linear combination of $\xi_\pm^{-i\tau}$ which is thus bounded, proving the sought $L^2$ boundedness.

If $\tau =0$, then $u$ is homogeneous of degree $-1$ in one dimension. You need $u$ to be odd for the $L^2$ boundedness to hold. Take for instance (still in one dimension) $u(x)=1/\vert x\vert$, obviously homogeneous with degree $-1$ and smooth on $\mathbb R^*$. The singular integral with kernel $1/\vert x-y\vert$ is not bounded on $L^2$, but the Hilbert transform with kernel $1/(x-y)$ is bounded on $L^2$ (with norm π).

Answer 2

By the Proposition 2.4.8. of L. Grafakos(GTM249), we have $\hat{u}\in C^{\infty}\left(\mathbb{R}^{n} \backslash\{0\}\right)$. It follows from $u$ is homogeneous of degree $-n+i\tau$ that $\hat{u}$ is a homogeneous distribution of degree $-i\tau$.Then, $$\hat{u}(\xi)=|\xi|^{-i\tau}\hat{u}(\frac{\xi}{|\xi|}),$$ which implies that $|\hat{u}(\xi)|=\sup_{|x|=1}|\hat{u}(x)|<\infty$, i.e. $\hat{u}\in L^{\infty}(\mathbb{R}^{n})$. Thus, $$\|u*f\|_{L^{2}}=\|\hat{u}\hat{f}\|_{L^{2}}\leq \|\hat{u}\|_{L^{\infty}}\|\hat{f}\|_{L^{2}}=\|\hat{u}\|_{L^{\infty}}\|f\|_{L^{2}}$$ for all $f \in L^{2}(\mathbb{R}^{n})$.