A relative Hofer estimate and the asymptotic Hofer-Lipschitz constant

Let $(M,\omega)$ be a symplectic manifold and $U\subseteq M$ an open subset. We study the natural inclusion of the compactly supported Hamiltonian group of $U$ in the compactly supported Hamiltonian group of $M$. The main result is an upper bound for this map in terms of the Hofer norms for $U$ and $M$. Applications are upper bounds on the asymptotic Hofer-Lipschitz constant and the relative Hofer diameter of $U$. The first bound is often sharp and the second one is often sharp up to a factor of 2.