Analysis of degenerate elliptic operators of Grušin type

Derek W. Robinson, Adam Sikora

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18 Citations (Scopus)


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 Hδ for some a1, a2 > 0 where Hδ=-∇x1·(c δ1,δ′1 (x1) ∇x1) - cδ2,δ′2 (x1) ∇x2 2. Here x1 ∈ Rn , x2 ∈ Rm and cδi,δ′i are positive measurable functions such that cδi,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 St = e-tH is conservative and its kernel K t satisfies bounds 0 ≤ K t(x; y) ≤ a (|B(x; t1/2)||B(y; t1/2)|) -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 languageEnglish
Pages (from-to)475-508
Number of pages34
JournalMathematische Zeitschrift
Issue number3
Publication statusPublished - Nov 2008
Externally publishedYes


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