Abstract:
Let S be the semigroup on L₂(Rd) generated by a degenerate elliptic operator, formally equal to −∑∂k ckl ∂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) ≥ μ I for all x ∈ Ω. We show that the operator PΩ St PΩ has a kernel satisfying Gaussian bounds and Gaussian Hölder bounds, where PΩ is the projection of L2(Rd) onto L2(Ω). Similar results are for the operators u → χ St(χ u), where χ ∈ C∞ b (Rd) and C(x) ≥ μI for all x ∈ supp χ.