Bilinear Calderón-Zygmund theory on product spaces

Abstract

We develop a wide general theory of bilinear bi-parameter singular integrals $T$. First, we prove a dyadic representation theorem starting from $T1$ assumptions and apply it to show many estimates, including $L^p\times L^q\to L^r$ estimates in the full natural range together with weighted estimates and mixed-norm estimates. Second, we develop commutator decompositions and show estimates in the full range for commutators and iterated commutators, like $[b_1,T{]}_1$ and $[b_2,[b_1,T{]}_1{]}_2$, where $b_1$ and $b_2$ are little BMO functions. Our proof method can be used to simplify and improve linear commutator proofs, even in the two-weight Bloom setting. We also prove commutator lower bounds by using and developing the recent median method.

v4: Edited combination of previous v3, arXiv:1804.06296 and arXiv:1806.10085 with a new title and some additional new results. In particular, this replaces arXiv:1804.06296 and arXiv:1806.10085. This version is submitted for publication. Now 55 pages

References in corpus (10)

• Multiple Vector Valued Inequalities via the Helicoidal Method
• Commutators of singular integrals revisited
• Weighted little bmo and two-weight inequalities for Journé commutators
• End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications
• The Sparse T1 Theorem
• The L^p-to-L^q boundedness of commutators with applications to the Jacobian operator
• Sparse Domination for Bi-Parameter Operators Using Square Functions
• Two-weight inequalities for multilinear commutators
• The unified theory for the necessity of bounded commutators and applications
• Dyadic representation and boundedness of non-homogeneous Calderón--Zygmund operators with mild kernel regularity

Cited by in corpus (10)

• End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications
• Off-diagonal estimates for bi-commutators
• Two-weight commutator estimates: general multi-parameter framework
• Modern singular integral theory with mild kernel regularity
• Some new weighted estimates on product spaces
• Genuinely multilinear weighted estimates for singular integrals in product spaces
• Commutators of bilinear bi-parameter singular integrals
• Bloom type upper bounds in the product BMO setting
• 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