Commutator estimates and Poisson bounds for Dirichlet-to-Neumann operators with variable coefficients: Commutator estimates and Poisson bounds...: A.F.M. ter Elst, E.M. Ouhabaz.

We consider the Dirichlet-to-Neumann operator N associated with a general elliptic operator A u = - ∑ k , l = 1 d ∂ k (c kl ∂ l u) + ∑ k = 1 d (c k ∂ k u - ∂ k (b k u)) + c 0 u ∈ D ′ (Ω) with possibly complex coefficients. We study three problems: (1) Boundedness on C ν and on L p of the commutator [ N , M g ] , where M g denotes the multiplication operator by a smooth function g. (2) Hölder and L p -bounds for the harmonic lifting associated with A . (3) Poisson bounds for the heat kernel of N . We solve these problems in the case where the coefficients are Hölder continuous and the underlying domain is bounded and of class C 1 + κ for some κ > 0 . For the Poisson bounds we assume in addition that the coefficients are real-valued. We also prove gradient estimates for the heat kernel and the Green function G of the elliptic operator with Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]