Some new weighted estimates on product spaces

Abstract

We complete our theory of weighted $L^p(w_1)\times L^q(w_2)\to L^r(w_1^{r/p}{w}_2^{r/q})$ estimates for bilinear bi-parameter Calderón--Zygmund operators under the assumption that $w_1\in A_p$ and $w_2\in A_q$ are bi-parameter weights. This is done by lifting a previous restriction on the class of singular integrals by extending a classical result of Muckenhoupt and Wheeden regarding weighted BMO spaces to the product BMO setting. We use this extension of the Muckenhoupt-Wheeden result also to generalise some two-weight commutator estimates from bi-parameter to multi-parameter. This gives a fully satisfactory Bloom type upper estimate for $[T_1,[T_2,\dots [b,T_k]]]$, where each $T_i$ can be a completely general multi-parameter Calderón--Zygmund operator.

arXiv commentv3: final version, incorporated referee comments, to appear in Indiana University Mathematics Journal, 20 pages

References in corpus (6)

• Commutators in the Two-Weight Setting
• Extrapolation for multilinear Muckenhoupt classes and applications to the bilinear Hilbert transform
• Quantitative estimates and extrapolation for multilinear weight classes
• End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications
• Bilinear Calderón-Zygmund theory on product spaces
• Two-weight commutator estimates: general multi-parameter framework

Cited by in corpus (1)

• Modern singular integral theory with mild kernel regularity