乘积空间上的粗糙奇异积分与Marcinkiewicz积分，以及乘积球面上的$H^1$刻画
Rough Singular Integrals and Marcinkiewicz Integrals on Product Spaces, with $H^1$ Characterizations on Product Spheres
- Zhejiang Normal University（浙江师范大学）
- University of Wisconsin–Milwaukee（威斯康星大学密尔沃基分校）
- Zhejiang University（浙江大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文在乘积空间上证明了粗糙奇异积分、极大截断及Marcinkiewicz积分的$L^p$有界性，并利用受限Riesz变换和旋转方法给出了乘积Hardy空间的等价刻画。
中文摘要 AI 辅助
我们证明了粗糙乘积奇异积分$T_\Omega$、其极大截断$T_\Omega^*$以及乘积Marcinkiewicz积分$\mu_\Omega$在$1<p<\infty$时的$L^p(\mathbb R^n\times\mathbb R^m)$有界性，假设$\Omega\in H^1(S^{n-1}\times S^{m-1})$满足分离消没条件。这些结果包含了$H^1(S^{n-1}\times S^{m-1})$中的核，它们超出了早期工作中所需的Orlicz类。我们还通过受限、共轭和谱Riesz变换，以及与谱和球Poisson延拓相关的径向极大函数、非切向极大函数和面积积分，获得了乘积Hardy空间的等价刻画。算子有界性利用受限Riesz变换和Calderón--Zygmund旋转方法证明。对于Marcinkiewicz积分，我们进一步建立了变换核的径向轮廓的可积界。
英文摘要
We prove the $L^p(\mathbb R^n\times\mathbb R^m)$ boundedness of the rough product singular integral $T_Ω$, its maximal truncation $T_Ω^*$, and the product Marcinkiewicz integral $μ_Ω$ for $1<p<\infty$, assuming that $Ω\in H^1(S^{n-1}\times S^{m-1})$ satisfies separate cancellation. These results include kernels in $H^1(S^{n-1}\times S^{m-1})$ 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.
