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

Abstract

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, $\Vert b{\Vert }_{\mathrm{bmo}({\mathbb{R}}^{n+m})}=1$ and $1<p,q\le \infty$ and $1/2<r<\infty$ satisfy $1/p+1/q=1/r$, then we have $$\Vert [b,T{]}_1(f_1,f_2){\Vert }_{L^r({\mathbb{R}}^{n+m})}\lesssim \Vert f_1{\Vert }_{L^p({\mathbb{R}}^{n+m})}\Vert f_2{\Vert }_{L^q({\mathbb{R}}^{n+m})}.$$ 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.

19 pages. arXiv admin note: text overlap with arXiv:1806.02742 and arXiv:1804.06296

References in corpus (2)

• Bilinear Calderón-Zygmund theory on product spaces
• Commutators of bilinear bi-parameter singular integrals