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

We consider the Dirichlet-to-Neumann operator $\mathcal{N}$ associated with a general elliptic operator $$\mathcal{A}u=-\sum _{k,l=1}^d{\mathrm{\partial }}_k(c_{kl}{\mathrm{\partial }}_{l}u)+\sum _{k=1}^d(c_k{\mathrm{\partial }}_{k}u-{\mathrm{\partial }}_k(b_{k}u))+c_{0}u\in {\mathcal{D}}^{\mathrm{\prime }}(\mathrm{\Omega })$$ with possibly complex coefficients. We study three problems: 1) Boundedness on $C^{\nu }$ and on $L_p$ of the commutator $[\mathcal{N},M_g]$, where $M_g$ denotes the multiplication operator by a smooth function $g$. 2) Hölder and $L_p$-bounds for the harmonic lifting associated with $\mathcal{A}$. 3) Poisson bounds for the heat kernel of $\mathcal{N}$. We solve these problems in the case where the coefficients are Hölder continuous and the underlying domain is bounded and of class $C^{1+\kappa }$ 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.