## Abstract

We examine the validity of the Poincaré inequality for degenerate, second-order, elliptic operatorsH in divergence form on L2(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 Hd is a generalized Grušin operator,(Formula presented.).Here (Formula presented.) and (Formula presented.) and(Formula presented.). We prove that the Poincaré inequality, formulated in terms of thegeometry corresponding to the control distance of H, is valid if n ≥ 2, or if n = 1 and δ_{1} ∨ δ′_{1} ∈ [0, 1/2〉 but it fails if n = 1 and δ_{1} ∨ δ′_{1} ∈ [1/2, 1 〉. The failure is caused by the leading term. If δ_{1} ∈ [1/2, 1〉, it is an effect of the local degeneracy {pipe}x1{pipe}^{2δ1}, but if δ1 ∈ [0, 1/2〉 and δ^{1} ∈ [1/2, 1〉, it is an effect of the growth at infinity of {pipe}x1{pipe}^{2δ′}1. If n = 1 and δ_{1} ∈ [1/2, 1〉, then the semigroup S generated by the Friedrichs' extensionof H is not ergodic. The subspaces x_{1} ≥ 0 and x_{1}≤ 0 are S-invariant, and the Poincaré inequality is validon each of these subspaces. If, however, n = 1, δ_{1} ∈ [0, 1/2〉 and δ′_{1} ∈ [1/2, 1〉, then the semigroup S isergodic, but the Poincaré inequality is only valid locally. Finally, we discuss the implication of these resultsfor the Gaussian and non-Gaussian behaviour of the semigroup S.

Original language | English |
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Pages (from-to) | 535-563 |

Number of pages | 29 |

Journal | Journal of Evolution Equations |

Volume | 14 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 2014 |