In this Lecture we consider a complete and -dimensional Riemannian manifold with non negative Ricci curvature. Our goal is to prove the following fundamental result, which is known as the volume doubling property.

**Theorem:*** There exists a constant such that for every and every one has *

Actually by suitably adapting the arguments given in this Lecture, the previous result can be extended to the case of negative Ricci curvature as follows:

**Theorem:*** Assume with . There exist positive constants such that for every and every one has
*

For simplicity, we show the arguments in the case and let the reader work out the arguments in the case .

This result can be obtained from geometric methods as a consequence of the Bishop-Gromov comparison theorem. The proof we give instead only relies on the previous methods and has the advantage to generalize to a much larger class of operators than Laplace-Beltrami on Riemannian manifolds.

The key heat kernel estimate that leads to the doubling property is the following uniform and scale invariant lower bound on the heat kernel measure of balls.

**Theorem:** *There exist an absolute constant , and , depending only on , such that
*

**Proof:** We first recall the following result that was proved in a previous Lecture: Let and . Given , which is bounded and such that is Lipschitz, we have

We choose ,and where will later be optimized. Noting that we presently have

we obtain the inequality

In what follows we consider a bounded function on such that almost everywhere on . For any we consider the function defined by

Notice that Jensen’s inequality gives and so we have We now apply the previous inequality to the function , obtaining

Keeping in mind that , we see that Using this observation in combination with the fact that

and switching notation from to , we infer

The latter inequality finally gives

We now optimize the right-hand side of the inequality with respect to . We notice explicitly that the maximum value of the right-hand side is attained at . We find therefore

We now integrate the inequality between and , obtaining

We infer then

Letting we conclude

At this point we let , and consider the function . Since we clearly have it follows that for every one has This gives the lower bound . To estimate the first term in the right-hand side of the latter inequality, we use the previous estimate which gives To make use of this estimate, we now choose , , obtaining . The conclusion follows then easily

We now turn to the proof of the volume doubling property. We first recall the following basic result which is a straightforward consequence of the Li Yau inequality.

**Proposition:*** Let be the heat kernel on . For every and every one has
*

We are now in position to prove the doubling.

From the semigroup property and the symmetry of the heat kernel we have for any and Consider now a function such that , on and outside . We thus have

If we take , and , we obtain

At this point we use the crucial previous theorem, which gives for some

Combining the latter inequality with the Harnack inequality, we obtain the following on-diagonal lower bound

Applying the Harnack inequality for every we find

Integration over gives

where we have used . Letting , we obtain from this the on-diagonal upper bound . We finally obtain

where we have used once more the Harnack inequality,

which gives .