Commutators of bilinear bi-parameter singular integrals

Abstract

We study the boundedness properties of commutators formed by $b$ and $T$, where $T$ is a bilinear bi-parameter singular integral satisfying natural $T1$ type conditions and $b$ is a little BMO function. For paraproduct free bilinear bi-parameter singular integrals $T$ we prove that $[b,T{]}_1:L^p({\mathbb{R}}^{n+m})\times L^q({\mathbb{R}}^{n+m})\to L^r({\mathbb{R}}^{n+m})$ in the full range $1<p,q\le \infty$, $1/2<r<\infty$ satisfying $1/p+1/q=1/r$. A special case is when $T$ is a bilinear bi-parameter multiplier. We also prove the corresponding Banach range result for all singular integrals satisfying the $T1$ type conditions. In doing so we simplify the corresponding linear proof. Lastly, we prove analogous results for iterated commutators.

37 pages

References in corpus (3)

• Multiple Vector Valued Inequalities via the Helicoidal Method
• Bilinear Calderón-Zygmund theory on product spaces
• Two-weight inequalities for multilinear commutators

Cited by in corpus (2)

• Quasi-Banach estimates of commutators of bilinear bi-parameter singular integrals: paraproducts
• Bloom type inequality for bi-parameter singular integrals: efficient proof and iterated commutators