Two-weight commutator estimates: general multi-parameter framework. (English) Zbl 1471.42029

Let \(T_i\) be an \(m_i\)-parameter Calderón-Zygmund operator, \(1\le i\le k\), and let \(b\) belong to a suitable weighted little product BMO space. The author considered the following two weight commutator estimates \[ [T_1, [T_2, \dots, [b, T_k]]]: L^p(\mu)\mapsto L^p(\lambda), \] where \(1<p<\infty\), \(\mu, \lambda\in A_p\) and \(\nu=\mu^{\frac 1p}\lambda^{-\frac 1p}\) is the Bloom weight. In other words, the author wanted to provide the multi-parameter analog of the recent work by K. Li et al. [J. Geom. Anal. 30, No. 3, 3181–3203 (2020; Zbl 1440.42063)].
The main result of the paper under review is that, if for all \(1\le i\le k\), either \(T_i\) is paraproduct free or \(m_i\le 2\), then \(\|[T_1, [T_2, \dots, [b, T_k]]]\|_{L^p(\mu)\to L^p(\lambda)}\) is controlled by suitable weighted little product BMO norm of \(b\).
In the recent work [E. Airta et al., “Some new weighted estimates on product spaces”, Preprint, arXiv:1910.12546], the restrictions on \(T_i\), i.e., either \(T_i\) is paraproduct free or \(m_i\le 2\), are removed.


42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B35 Function spaces arising in harmonic analysis


Zbl 1440.42063
Full Text: DOI arXiv Euclid


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