Partial Gaussian Bounds for Degenerate Differential Operators

Abstract

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 ckl are real bounded measurable and the matrix C(x) = (ckl(x)) is symmetric and positive semi-definite for all x ∈ Rd. Let Ω ⊂ Rd be a bounded Lipschitz domain and μ > 0. Suppose that C(x) ≥ μI for all x ∈ Ω. We show that the operator PΩStPΩ has a kernel satisfying Gaussian bounds and Gaussian Holder bounds, where PΩ is the projection of \(L_2({{\bf R}}^d)\) onto L2(Ω). Similar results are for the operators u ↦ χSt (χu), where \(\chi \in C_{\rm b}^\infty({{\bf R}}^d)\) and C(x) ≥ μI for all \(x \in {\mathop{\rm supp}} \chi\).