Riesz transform on manifols and Poincaré inequalities

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Riesz transform on manifolds and heat kernel regularity (2004, background): “In a subsequent paper [5], the two first-named authors of the present article will extend these ideas to the manifold setting and prove actually that there always is some p0 > 2 for which our Theorem 1.”
Gradient estimates for heat kernels and harmonic functions (2017, background): “For further information we refer to [6, 15] and references therein.”
Second order elliptic operators with complex bounded measurable coefficients in L p , Sobolev and Hardy spaces (2010, background): “The argument here resonates with that in [7], Section 1.”
Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights (2006, methodology): “See also [5] for an extension to Riemannian manifolds.”