The goal of this lecture is to prove that if a diffusion operator L is elliptic, then the semigroup it generates admits a smooth kernel. As a consequence, the semigroup generated by an elliptic diffusion operator is regularizing in the sense that it transforms any function into a smooth function. The key point is the following estimate that can be proved by using the theory of Sobolev spaces.
Proposition: Let be an elliptic diffusion operator with smooth coefficients on which is symmetric with respect to a Borel measure . Let such that , for some positive integer . If , then is a continuous function, moreover, for any bounded open set , and any compact set , there exists a positive constant (independent of ) such that More generally, if for some non negative integer , then and for any bounded open set , and any compact set , there exists a positive constant (independent of ) such that
We can explain by simple computations on the Laplace operator how the comes into the play in the above proposition. Let be a smooth and rapidly decreasing function. By the inverse Fourier transform formula so that by Cauchy-Schwarz inequality we may bound
by only when that is if . We are now in position to prove the following regularization estimate for the semigroup associated with an elliptic operator.
Proposition: Let be an elliptic 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 .
- 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 previous proposition, 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 an elliptic 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