A Note on “Riesz Transform for $1 \le p \le 2$ Without Gaussian Heat Kernel Bound”

Abstract

Consider the direct product manifold $M_1 \times \cdots \times M_n$, where $M_i$ ($1 \le i \le n$) are connected complete non-compact Riemannian manifolds satisfying the volume doubling property and Gaussian or sub-Gaussian estimates for the heat kernel. We obtain weak type (1, 1) (so $L^p$-boundedness with $1< p < 2$) for the Riesz transform. As a consequence, we find that neither heat kernel Gaussian estimates nor sub-Gaussian estimates are necessary for weak (1, 1) property of Riesz transform.