Let $T : S(R) \rightarrow C_b(R)$ be a continuous linear operator which maps Schwartz functions to bounded continuous, and which commutes with translations and dilations. Show that T is a linear multiple of the Hilbert transform and the identity.
(Hint: first establish the existence of a tempered distribution $K \in S(R)^*$ such that
$$Tf(x) = \left<K,\text{Trans}_{-x}f \right> \ \ \ \text{for all} \ \ f \in S(R),$$
and such that $K$ is homogeneous of degree $−1$. Restrict the action of K to Schwartz functions on $R_+$ or $R_-$ and show that $K$ is a constant multiple of $1/x$ on these functions.) (The claim is also true if we relax $C_b(R)$ to $S(R)^*$, the space of tempered distributions, but this is a little trickier to prove.)
Comment: My question is: how to show this K is principal value tempered distribution?