I'm reading the Petersen's book "Riemannian Geometry". My goal is to learn the proof of the Cheeger-Gromoll Splitting Theorem. It's quite complex and made by a lot of step. Now I'm trying to understand the proof of the Lemma 42 of section 3.3 (page 284).
The statement is:
LEMMA: Let $r(x) = d(x,p)$, with $p \in (M,g)$. If $Ric \ge 0$, then $$ \Delta r(x) \le \frac{n-1}{r(x)} \qquad \text{for all }x \in M.$$
The first sentence in the proof is "We know that the result is true whenever $r$ is smooth." Why? I can't see it.
I think Petersen is referring to Lemma 34 on page 268, where he proves a more general Laplacian estimate under arbitrary lower Ricci bounds. Although the statement of that lemma doesn't mention it, in the proof it's evident that he's assuming $r$ is smooth. Thus the point of Lemma 42 on page 284 is to extend that result to the more general case in which $r$ is merely Lipschitz continuous.