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

Abstract

Let $\Omega$ be a bounded domain of $\mathbb{R}^{n+1}$ with $n \ge 1$. We assume that the boundary $\Gamma$ of $\Omega$ 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 $L^p(\Gamma)\to L^q(\Gamma)$ off-diagonal bounds of the form$$ \| 1_F e^{-t N_0} 1_E f \|_q \lesssim (t \wedge 1)^{\frac{n}{q}-\frac{n}{p}} \left( 1 + \frac{dist(E,F)}{t} \right)^{-1} \| 1_E f \|_p$$for all measurable subsets $E$ and $F$ of $\Gamma$. If $\Gamma$ is $C^{1+ \kappa}$ for some $\kappa > 0$ and $m=1$, we obtain a sharp estimate in the sense that $ \left( 1 + \frac{dist(E,F)}{t} \right)^{-1}$ can be replaced by$ \left( 1 + \frac{dist(E,F)}{t} \right)^{-(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 $L^p(\Gamma)$ for all $p \in (1, \infty)$. In addition, the corresponding evolution problem has $L^q(L^p)$-maximal regularity.