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

We consider the Dirichlet-to-Neumann operator ${\cal N}$ associated with a general elliptic operator $${\cal A} u = - \sum_{k,l=1}^d \partial_k (c_{kl}\, \partial_l u) + \sum_{k=1}^d \Big( c_k\, \partial_k u - \partial_k (b_k\, u) \Big) +c_0\, u \in {\cal D}'(\Omega)$$ with possibly complex coefficients. We study three problems: 1) Boundedness on $C^\nu$ and on $L_p$ of the commutator $[{\cal N}, M_g]$, where $M_g$ denotes the multiplication operator by a smooth function $g$. 2) H\"older and $L_p$-bounds for the harmonic lifting associated with ${\cal A}$. 3) Poisson bounds for the heat kernel of ${\cal N}$. We solve these problems in the case where the coefficients are H\"older continuous and the underlying domain is bounded and of class $C^{1+\kappa}$ for some $\kappa > 0$. 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.