In this Lecture, we use the local regularity theory of subelliptic operators, to prove the existence of heat kernels.

**Proposition:*** Let be a locally subelliptic diffusion operator with smooth coefficients that is essentially self-adjoint with respect to a measure . Denote by the corresponding semigroup on .*

- If is a compact set of , there exists a positive constant such that for , where is the smallest integer larger than and .
- For , the function is smooth on .

**Proof:** Let us first observe that from the spectral theorem that if then for every , and Now, let be a compact set of . From the proposition in, there exists therefore a positive constant such that Since it is immediately checked that the bound easily follows. We now turn to the second part. Let . First, we fix . As above, from the spectral theorem, for every , , for any bounded open set . By hypoellipticity of , we deduce therefore that is a smooth function.

Next, we prove joint continuity in the variables . It is enough to prove that if and if is a compact set in , From the previous proposition, there exists a positive constant such that Now, again from the spectral theorem, it is checked that This gives the expected joint continuity in . The joint smoothness in is a consequence of the second part of the previous proposition and the details are let to the reader

**Remark:*** If the bound uniformly holds on , that is if then the semigroup is said to be ultracontractive.*

**Exercise:*** Let be an elliptic diffusion operator with smooth coefficients that is essentially self-adjoint with respect to a measure . Let be a multi-index. If is a compact set of , show that there exists a positive constant such that for , where is the smallest integer larger than .*

We are now in position to prove the following fundamental theorem:

**Theorem:*** Let be a locally subelliptic and essentially self-adjoint diffusion operator. Denote by the corresponding semigroup on . There is a smooth function , , such that for every and , The function is called the heat kernel associated to . It satisfies furthermore:*

- (Symmetry) ;
- (Chapman-Kolmogorov relation) .

**Proof:** Let and . From the previous proposition, the linear form is continuous on , therefore from the Riesz representation theorem, there is a function , such that for , From the fact that is self-adjoint on , we easily deduce the symmetry property And the Chapman-Kolmogorov relation stems from the semigroup property . Finally, from the previous proposition the map is smooth on for the weak topology of . This implies that it is also smooth on for the norm topology. Since, from the Chapman-Kolmogorov relation we conclude that is smooth on