Continuous operators from spaces of Lipschitz functions

We study the existence of continuous (linear) operators from the Banach spaces Lip₀(M) of Lipschitz functions on infinite metric spaces M vanishing at a distinguished point and from their predual spaces F(M) onto certain Banach spaces, including $C(K)$-spaces and the spaces c₀ and ₁. For pairs of spaces Lip₀(M) and $C(K)$ we prove that if they are endowed with topologies weaker than the norm topology, then usually no continuous (linear or not) surjection exists between those spaces. We show that, given a Banach space E, there exists a continuous operator from a Lipschitz-free space F(M) onto E if and only if F(M) contains a subset homeomorphic to E if and only if d(M)≥ d(E). We obtain a new characterization of the Schur property for spaces F(M): a space F(M) has the Schur property if and only if for every discrete metric space N with cardinality $d(M)$ the spaces F(M) and F(N) are weakly sequentially homeomorphic. It is also showed that if a metric space M contains a bilipschitz copy of the unit sphere Sc₀ of the space c₀, then Lip₀(M) admits a continuous operator onto ₁ and hence onto c₀. We provide several conditions for a space M implying that Lip₀(M) is not a Grothendieck space.