A Note on “Riesz Transform for $$1 \le p \le 2$$ 1 ≤ p ≤ 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.