𝐻^{𝑝}- and 𝐿^{𝑝}-variants of multiparameter Calderón-Zygmund theory

Carbery, Anthony; Seeger, Andreas

doi:10.1090/S0002-9947-1992-1072104-4

$H^ p$- and $L^ p$-variants of multiparameter Calderón-Zygmund theory
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Trans. Amer. Math. Soc. 334 (1992), 719-747 Request permission

Abstract:

We consider Calderón-Zygmund operators on product domains. Under certain weak conditions on the kernel a singular integral operator can be proved to be bounded on ${H^p}(\mathbb {R} \times \mathbb {R} \times \cdots \times \mathbb {R}), 0 < p \leq 1$, if its behaviour on ${L^2}$ and on certain scalar-valued and vector-valued rectangle atoms is known. Another result concerns an extension of the authors’ results on ${L^p}$-variants of Calderón-Zygmund theory [1,23] to the product-domain-setting. As an application, one obtains estimates for Fourier multipliers and pseudo-differential operators.

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