Free Banach spaces and extension of Lipschitz maps

Let XX be a metric space. We study the free Banach space B(X)B(X) over XX, that is a predual space of the Banach space of all Lipschitz functions on XX which preserve a marked point θ∈Xθ∈X. Some applications to the extension theory of Lipschitz and two-Lipschitz functions are obtained.