Off-diagonal bounds for the Dirichlet-to-Neumann operator

Let $\mathrm{\Omega }$ be a bounded domain of ${\mathbb{R}}^{n+1}$ with $n\ge 1$. We assume that the boundary $\mathrm{\Gamma }$ of $\mathrm{\Omega }$ is Lipschitz. Consider the Dirichlet-to-Neumann operator $N_0$ associated with a system in divergence form of size $m$ with real symmetric and H''older continuous coefficients. We prove $L^p(\mathrm{\Gamma })\to L^q(\mathrm{\Gamma })$ off-diagonal bounds of the form$$\mathrm{\parallel }{1}_F{e}^{-t{N}_0}{1}_{E}f{\mathrm{\parallel }}_q\lesssim (t\wedge 1{)}^{\frac{n}{q}-\frac{n}{p}}{(1+\frac{dist(E,F)}{t})}^{-1}\mathrm{\parallel }{1}_{E}f{\mathrm{\parallel }}_p$$for all measurable subsets $E$ and $F$ of $\mathrm{\Gamma }$. If $\mathrm{\Gamma }$ is $C^{1+\kappa }$ for some $\kappa >0$ and $m=1$, we obtain a sharp estimate in the sense that ${(1+\frac{dist(E,F)}{t})}^{-1}$ can be replaced by${(1+\frac{dist(E,F)}{t})}^{-(1+\frac{n}{p}-\frac{n}{q})}$. Such bounds are also valid for complex time. For $n=1$, we apply our off-diagonal bounds to prove that the Dirichlet-to-Neumann operator associated with a system generates an analytic semigroup on $L^p(\mathrm{\Gamma })$ for all $p\in (1,\mathrm{\infty })$. In addition, the corresponding evolution problem has $L^q(L^p)$-maximal regularity.