Partial Gaussian Bounds for Degenerate Differential Operators

Let S be the semigroup on \(L_2({{\bf R}}^d)\) generated by a degenerate elliptic operator, formally equal to \(- \sum \partial_k \, c_{kl} \, \partial_l\), where the coefficients c _kl are real bounded measurable and the matrix C(x)?=?(c _kl (x)) is symmetric and positive semi-definite for all x?∈?R ^d . Let Ω???R ^d be a bounded Lipschitz domain and μ?>?0. Suppose that C(x)?≥?μ I for all x?∈?Ω. We show that the operator P _Ω S _t P _Ω has a kernel satisfying Gaussian bounds and Gaussian Hölder bounds, where P _Ω is the projection of \(L_2({{\bf R}}^d)\) onto L ₂(Ω). Similar results are for the operators u ? χ S _t (χ u), where \(\chi \in C_{\rm b}^\infty({{\bf R}}^d)\) and C(x)?≥?μI for all \(x \in {\mathop{\rm supp}} \chi\).