The limitations of the Poincaré inequality for Grušin type operators

We examine the validity of the Poincaré inequality for degenerate, second-order, elliptic operatorsH in divergence form on L2(Rⁿ × R^m). We assume the coefficients are real symmetric and a₁H_δ ≥ H ≥ a₂H_δ for some a₁, a₂ > 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 δ₁ ∨ δ′₁ ∈ [0, 1/2〉 but it fails if n = 1 and δ₁ ∨ δ′₁ ∈ [1/2, 1 〉. The failure is caused by the leading term. If δ₁ ∈ [1/2, 1〉, it is an effect of the local degeneracy {pipe}x1{pipe}^2δ1, but if δ1 ∈ [0, 1/2〉 and δ¹ ∈ [1/2, 1〉, it is an effect of the growth at infinity of {pipe}x1{pipe}^2δ′1. If n = 1 and δ₁ ∈ [1/2, 1〉, then the semigroup S generated by the Friedrichs' extensionof H is not ergodic. The subspaces x₁ ≥ 0 and x₁≤ 0 are S-invariant, and the Poincaré inequality is validon each of these subspaces. If, however, n = 1, δ₁ ∈ [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.