integrability of Hilbert transform of a function

Question

The problem is :

Let $\varphi \in \mathcal S(\mathbb R)$ (Schwartz space) with $\int \varphi \ dx = 0$. Then the Hilbert transform of $\varphi$ belongs to $L^1(\mathbb R)$.

I believe this helps $\lim_{|x| \rightarrow \infty} x H(\varphi) (x) = 0$.

But i dont know what to do..

someone can give me a hint ?

Thanks in advance