## Abstract

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 ≥ a_{2} H_{δ} for some a_{1}, a_{2} > 0 where H_{δ}=-∇_{x1}·(c _{δ1,δ′1} (x_{1}) ∇_{x1}) - c_{δ2,δ′2} (x_{1}) ∇_{x2} ^{2}. Here x_{1} ∈ R^{n} , x_{2} ∈ 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 δ_{1}, δ′_{1} ∈ [0, 1〉 and δ_{2}, 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.

Original language | English |
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Pages (from-to) | 475-508 |

Number of pages | 34 |

Journal | Mathematische Zeitschrift |

Volume | 260 |

Issue number | 3 |

DOIs | |

Publication status | Published - Nov 2008 |

Externally published | Yes |