Quasi-Banach estimates of commutators of bilinear bi-parameter singular integrals: paraproducts

We complete our boundedness theory of commutators of bilinear bi-parameter singular integrals by establishing the following result. If $T$ is a bilinear bi-parameter singular integral satisfying suitable $T1$ type assumptions, $\|b\|_{\operatorname{bmo}(\mathbb{R}^{n+m})} = 1$ and $1 < p, q \le \infty$ and $1/2 < r < \infty$ satisfy $1/p+1/q = 1/r$, then we have $$ \|[b, T]_1(f_1, f_2)\|_{L^r(\mathbb{R}^{n+m})} \lesssim \|f_1\|_{L^p(\mathbb{R}^{n+m})} \|f_2\|_{L^q(\mathbb{R}^{n+m})}. $$ Previously the range $r \le 1$ was proved only in the paraproduct free situation. The main novelty lies in the treatment of the so called partial paraproducts.