Uniqueness in 1D transport equation .

Question

I am trying to prove uniqueness in 1D transport equation $$ \theta_t-(H\theta)\theta_x=0\\ \theta(\cdot,0)=\theta_0 $$ being $Hf$ the Hilbert transform of $f$, $x,t\in\mathbb{R}$. To do that, I consider two solutions of the equation with the same initial data. The function $\hat{\theta}=\theta_1-\theta_2$ satisfies the equation $$ \hat{\theta}_t=(H\theta_2)\hat{\theta}_x+(H\hat{\theta})(\theta_1)_x. $$ Then, I multiply by $\hat{\theta}$ in $L^2$, but I do not know how to bound the term $$ \left((H\hat{\theta})(\theta_1)_x,\hat{\theta}\right). $$ Then, I would apply Gronwall lemma to obtain the uniqueness. Thanks.