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arXiv · 1910.12546

Some new weighted estimates on product spaces

Emil Airta·Kangwei Li·Henri Martikainen·Emil Vuorinen
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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 Calder\'on--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 $[T_1, [T_2, \ldots [b, T_k]]]$, where each $T_i$ can be a completely general multi-parameter Calder\'on--Zygmund operator.

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Emil Airta, Kangwei Li, Henri Martikainen, Emil Vuorinen. 2019-10-28. Some new weighted estimates on product spaces. https://arxiv.org/abs/1910.12546

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