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
Here x 1 ∈ R n, x 2 ∈ R m and \({{c_{\delta_{i}},\delta'_{i}}}\) are positive measurable functions such that \({c_{\delta_{i,}} \delta'_{i}(x)}\) behaves like \({|x|^{\delta_{i}}}\) as x → 0 and \({|x|^{\delta'_{i}}}\) as \(x \to \infty\) with \(\delta_1, \delta'_1 \in [0, 1 \rangle\) and \(\delta_2,\delta'_2 \geq 0\) . Our principal results state that the submarkovian semigroup \(S_t = e^{-tH}\) is conservative and its kernel K t satisfies bounds
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.
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Robinson, D.W., Sikora, A. Analysis of degenerate elliptic operators of Grušin type. Math. Z. 260, 475–508 (2008). https://doi.org/10.1007/s00209-007-0284-3
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DOI: https://doi.org/10.1007/s00209-007-0284-3