| Abstract: |
| We prove mean value formulas for classical solutions to second order linear differential equations in the form
$$
\p_t u = \sum_{i,j=1}^m X_i (a_{ij} X_j u) + X_0 u + f,
$$
where $A = (a_{ij})_{i,j=1, \dots,m}$ is a bounded, symmetric and uniformly positive matrix with $C^1$ coefficients under the assumption that the operator $\displaystyle\sum_{j=1}^m X_j^2 + X_0 - \p_t$ is hypoelliptic and the vector fields $X_1, \dots, X_m$ and $X_{m+1} :=X_0 - \p_t$ are invariant with respect to a suitable homogeneous Lie group. Our results apply e.g. to degenerate Kolmogorov operators and parabolic equations on Carnot groups $\displaystyle \p_t u = \sum_{i,j=1}^m X_i (a_{ij} X_j u) + f$.
An elementary proof of the strong maximum principle is obtained as an application of the mean value formulas. |
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