At present, no general representation theorem for duals of spaces of Lipschitz functions over a pointed metric space is known. A useful substitute is provided by a construction due to K. de Leeuw from the 1960s, which allows us to view functionals on spaces of Lipschitz functions as Radon measures integrating the incremental quotients of functions. The main drawback of such a representation is that the representing measures are not unique.
We will focus on functionals from the canonical predual of the space of Lipschitz functions, the Lipschitz-free space. We will discuss the existence of "nice" representing measures for such functionals and present some applications to the isometric theory of Lipschitz-free spaces. In particular, we will show that a Choquet-like theory for De Leeuw representations, recently developed by R. J. Smith, leads to an "inner regularity" result for elements of Lipschitz-free spaces and the characterisation of the extreme points of their unit balls.
The talk will be based on joint work with Ramón J. Aliaga (Universitat Politècnica de València) and Richard J. Smith (University College Dublin).