Positivity and ellipticity

We consider partial differential operators H = − div(C∇) in divergence form on R with a positive-semidefinite, symmetric, matrix C of real L∞-coefficients. First, we prove that one can define H as a selfadjoint operator on L2(R ) such that the corresponding semigroup extends as a positive, contraction semigroup to all the Lp-spaces. Secondly, we establish that H is strongly elliptic if and only if the distribution kernel of the semigroup satisfies appropriate small time lower bounds. Thirdly, we analyze degenerate operators satisfying the subellipticity condition H ≥ μ∆1−γ − ν I for some μ > 0, ν ≥ 0 and γ ∈ [0, 1〉, where ∆ denotes the usual Laplacian, and derive large time Gaussian bounds from a local condition of strict positivity.