*preprint*

**Inserted:** 11 sep 2023

**Last Updated:** 11 sep 2023

**Year:** 2023

**Abstract:**

For $1<p<\infty$, we prove the $L^p$-boundedness of the Riesz transform operators on metric measure spaces with Riemannian Ricci curvature bounded from below, without any restriction on their dimension. This large class of spaces include e.g. that of Hilbert spaces endowed with a log-concave probability measure. As a consequence, we extend the range of the validity of the Lusin-type approximation of Sobolev by Lipschitz functions, previously obtained by L. Ambrosio, E. Bru\`e and the third author in the quadratic case, i.e. $p=2$. The proofs are analytic and rely on computations on an explicit Bellman function.