Commutator estimates and Poisson bounds for Dirichlet-to-Neumann operators

We consider the Dirichlet-to-Neumann operator N associated with a general elliptic operator \ A u = - ₊, ₋=₁ᵈ ₖ (c₊₋\, ₗ u) + ₊=₁ᵈ (cₖ\, ₖ u - ₖ (bₖ\, u) ) +c₀\, u D' () \ with possibly complex coefficients. We study three problems: 1) Boundedness on C^ and on Lₚ of the commutator N, Mg, where Mg denotes the multiplication operator by a smooth function g. 2) H\"older and Lₚ-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 H\"older continuous and the underlying domain is bounded and of class C^1+ for some > 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.