Commutators of linear and bilinear Hilbert transforms

Abstract:

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 $\operatorname {sign}(\alpha _1)=\operatorname {sign}(\alpha _2)$. It is also shown that, for $\alpha \le 1$ the commutator $[H_{\alpha ,f},H]$ is bounded on $L^p$ for $1<p<\infty$ if and only if $f\in BMO$, where $H_{\alpha ,f}(g)=H_\alpha (f,g)$.