Injectivity of Lipschitz Operators

ny Lipschitz map f:M → N between metric spaces can be “linearised” in such a way that it becomes a bounded linear operator f:ℱ(M) →ℱ(N) between the Lipschitz-free spaces over M and N . The purpose of this note is to explore the connections between the injectivity of f and the injectivity of f . While it is obvious that if f is injective then so is f , the converse is less clear. Indeed, we pin down some cases where this implication does not hold but we also prove that, for some classes of metric spaces M , any injective Lipschitz map f:M → N (for any N ) admits an injective linearisation. Along our way, we study how Lipschitz maps carry the support of elements in free spaces and also we provide stronger conditions on f which ensure that f is injective.