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

Source 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 ( R × R × ⋯ × R ) , 0 > p ≤ 1 {H^p}(\mathbb {R} \times \mathbb {R} \times \cdots \times \mathbb {R}), 0 > p \leq 1 , if its behaviour on L 2 {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 {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.