Riesz transform under perturbations via heat kernel regularity

Let $M$ be a complete non-compact Riemannian manifold. In this paper, we derive sufficient conditions on metric perturbation for stability of $L^p$-boundedness of the Riesz transform, $pin (2,infty)$. We also provide counter-examples regarding in-stability for $L^p$-boundedness of Riesz transform.