In this paper we introduce atomic Hardy spaces on the product domain $S^{n - 1} \times S^{m - 1}$ and prove that rough singular integral operators with Hardy space function kernels are $L^{p}$ bounded on $ℝ^{n} \times ℝ^{m}$ . This is an extension of some well known results.

Maximal singular integrals on product homogeneous groups

Yong Ding; Shuichi Sato — 2014

Studia Mathematica

We prove $L^{p}$ boundedness for p ∈ (1,∞) of maximal singular integral operators with rough kernels on product homogeneous groups under a sharp integrability condition of the kernels.

Littlewood-Paley g-functions with rough kernels on homogeneous groups

Yong Ding; Xinfeng Wu — 2009

Studia Mathematica

Let 𝔾 be a homogeneousgroup on ℝⁿ whose multiplication and inverse operations are polynomial maps. In 1999, T. Tao proved that the singular integral operator with Llog⁺L function kernel on ≫ is both of type (p,p) and of weak type (1,1). In this paper, the same results are proved for the Littlewood-Paley g-functions on 𝔾

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