Analysis of degenerate elliptic operators of Grušin type

We analyse degenerate, second-order, elliptic operators H in divergence form on L ₂(R ⁿ × R ^m ). We assume the coefficients are real symmetric and a ₁ H _δ ≥ H ≥ a₂ H_δ for some a₁, a₂ > 0 where H_δ=-∇_x1·(c _δ1,δ′1 (x₁) ∇_x1) - c_δ2,δ′2 (x₁) ∇_x2 ². Here x₁ ∈ Rⁿ , x₂ ∈ R^m and c_δi,δ′i are positive measurable functions such that c_δi,δ′i (x) behaves like |x|^δi as x → 0 and |x|^δi as x → ∞ with δ₁, δ′₁ ∈ [0, 1〉 and δ₂, δ′₂ ≥ 0. Our principal results state that the submarkovian semigroup S_t = e^-tH is conservative and its kernel K _t satisfies bounds 0 ≤ K _t(x; y) ≤ a (|B(x; t^1/2)||B(y; t^1/2)|) ^-1/2 where |B(x; r)| denotes the volume of the ball B(x; r) centred at x with radius r measured with respect to the Riemannian distance associated with H. The proofs depend on detailed subelliptic estimations on H, a precise characterization of the Riemannian distance and the corresponding volumes and wave equation techniques which exploit the finite speed of propagation. We discuss further implications of these bounds and give explicit examples that show the kernel is not necessarily strictly positive, nor continuous.