We consider the Dirichlet-to-Neumann operator \({\mathcal {N}}\) associated with a general elliptic operator \(\begin{aligned} {\mathcal {A}}u = - \sum _{k,l=1}^d \partial _k (c_{kl}\, \partial _l u) + \sum _{k=1}^d \Bigg ( c_k\, \partial _k u - \partial _k (b_k\, u) \Bigg ) +c_0\, u \in {\mathcal {D}}'(\Omega ) \end{aligned}\) 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.
