Multi-parameter estimates via operator-valued shifts

Abstract

We prove new results for multi-parameter singular integrals. For example, we prove that bi-parameter singular integrals in ${\mathbb{R}}^{n+m}$ satisfying natural $T1$ type conditions map $L^q({\mathbb{R}}^n;L^p({\mathbb{R}}^m;E))$ to $L^q({\mathbb{R}}^n;L^p({\mathbb{R}}^m;E))$ for all $p,q\in (1,\infty )$ and UMD function lattices $E$. This result is shown to hold even in the $\mathcal{R}$-boundedness sense for all suitable families of bi-parameter singular integrals. On the technique side we demonstrate how many dyadic multi-parameter operators can be bounded by using, and further developing, the theory of operator-valued dyadic shifts. Even in the scalar-valued case this is an efficient way to bound the various so called partial paraproducts, which are key operators appearing in the multi-parameter representation theorems. Our proofs also entail verifying the $\mathcal{R}$-boundedness of various families of multi-parameter paraproducts.

41 pages

References in corpus (3)

• Weighted little bmo and two-weight inequalities for Journé commutators
• Sparse Domination for Bi-Parameter Operators Using Square Functions
• Rescaled extrapolation for vector-valued functions

Cited by in corpus (1)

• Off-diagonal estimates for bi-commutators