Commutators of linear and bilinear Hilbert transforms

Let $\alpha \in \mathbb{R}$, and let $H_\alpha(f,g)(x)=\frac{1}{\pi} p.v. \int f(x-t)g(x-\alpha t)\frac{dt}{t}$ and $Hf(x)= \frac{1}{\pi} p.v.\int f(x-t)\frac{dt}{t}$ denote the bilinear and linear Hilbert transforms, respectively. It is proved that, for $1<p<\infty$ and $\alpha_1\ne\alpha_2$, $H_{\alpha_1}-H_{\alpha_2}$ maps $L^p\times BMO$ into $L^{p}$ and it maps $BMO \times L^p$ into $L^{p}$ if and only if ${sign}(\alpha_1)={sign}(\alpha_2)$. It is also shown that, for $\alpha\le1$the commutator $[H_{\alpha,f},H]$ is bounded on $L^p$ for $1<p<\infty$ if and only if $f\in BMO$, where $H_{\alpha,f}(g)=H_\alpha(f,g)$.