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Rough Singular Integrals and Marcinkiewicz Integrals on Product Spaces, with H1 Characterizations on Product Spheres

  • Jiecheng Chen
  • Dashan Fan
  • Weitao Hu
  • Meng Wang
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Abstract

We prove the Lp(ℝn×ℝm) boundedness of the rough product singular integral TΩ, its maximal truncation TΩ*, and the product Marcinkiewicz integral μΩ for 1<p<∞, assuming that Ω∈ H1(Sn−1× Sm−1) satisfies separate cancellation. These results include kernels in H1(Sn−1× Sm−1) that lie outside the Orlicz classes required in earlier work. We also obtain equivalent characterizations of the product Hardy space in terms of restricted, conjugate, and spectral Riesz transforms, together with radial maximal functions, non-tangential maximal functions, and area integrals associated with the spectral and ball Poisson extensions. The operator bounds are proved using restricted Riesz transforms and the Calderón--Zygmund rotation method. For the Marcinkiewicz integral, we further establish integrable bounds for the radial profiles of the transformed kernels.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026rough,
  title = {{Rough Singular Integrals and Marcinkiewicz Integrals on Product Spaces, with $H^1$ Characterizations on Product Spheres}},
  author = {Jiecheng Chen and Dashan Fan and Weitao Hu and Meng Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.33696},
  eprint = {2609.33696},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33696},
}
RIS
TY  - UNPB
TI  - Rough Singular Integrals and Marcinkiewicz Integrals on Product Spaces, with H^1 Characterizations on Product Spheres
AU  - Jiecheng Chen
AU  - Dashan Fan
AU  - Weitao Hu
AU  - Meng Wang
PY  - 2026
DA  - 2026/09/27
PB  - arXiv
DO  - 10.48550/arxiv.2609.33696
UR  - https://arxiv.org/abs/2609.33696
AB  - We prove the L^p(ℝ^n×ℝ^m) boundedness of the rough product singular integral T_Ω, its maximal truncation T_Ω^*, and the product Marcinkiewicz integral μ_Ω for 1<p<∞, assuming that Ω∈ H^1(S^n−1× S^m−1) satisfies separate cancellation. These results include kernels in H^1(S^n−1× S^m−1) that lie outside the Orlicz classes required in earlier work. We also obtain equivalent characterizations of the product Hardy space in terms of restricted, conjugate, and spectral Riesz transforms, together with radial maximal functions, non-tangential maximal functions, and area integrals associated with the spectral and ball Poisson extensions. The operator bounds are proved using restricted Riesz transforms and the Calderón--Zygmund rotation method. For the Marcinkiewicz integral, we further establish integrable bounds for the radial profiles of the transformed kernels.
ER  -