Some new weighted estimates on product spaces

Emil Airta; Kangwei Li; Henri Martikainen; Emil Vuorinen

paper

Some new weighted estimates on product spaces

arXiv:1910.12546

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.

v3: final version, incorporated referee comments, to appear in Indiana University Mathematics Journal, 20 pages