Let $M$ be a doubling Riemannian manifold. Assume that $/Delta$ is the Laplace-Beltrami operator on $M$.  We also assume that $/Delta$ generates a semigroup with Gaussian upper bound. Then the Riesz transform $T = /nabla /Delta^{-1/2}$ (where $/nabla$ is the Riemannian gradient) is bounded on $L^2(M)$ and its kernel is non-smooth so that $T$ does not belong to the class of Calder/'on-Zygmund operators. we will show that T$ is of weak type (1,1), hence bounded on $L^p(M)$ for $1 < p /le 2$.

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