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

In this paper, we are concerned with the Riesz transform on the direct product manifold ℍⁿ × M, where ℍⁿ 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 et al. (2016).