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arXiv · 2304.04335

Weak type $(1, 1)$ of Riesz transform on some direct product manifolds with exponential volume growth

Hong-Quan Li·Jie-Xiang Zhu
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Abstract

In this paper we are concerned with the Riesz transform on the direct product manifold ${\mathbb{H}}^n \times M$, where ${\mathbb{H}}^n$ is the $n$-dimensional real hyperbolic space and $M$ is a connected complete non-compact Riemannian manifold satisfying the volume doubling property and generalized Gaussian or sub-Gaussian upper estimates for the heat kernel. We establish its weak type $(1,1)$ property. In addition, we obtain the weak type $(1, 1)$ of the heat maximal operator in the same setting. Our arguments also work for a large class of direct product manifolds with exponential volume growth. Particularly, we provide a simpler proof of weak type $(1,1)$ boundedness of some operators considered in the work of Li, Sj\"ogren and Wu [27].

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BibTeXRIS

Hong-Quan Li, Jie-Xiang Zhu. 2023-04-10. Weak type $(1, 1)$ of Riesz transform on some direct product manifolds with exponential volume growth. https://arxiv.org/abs/2304.04335

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