Connecting measures by means of branched transportation networks at finite cost

We study the couples of finite Borel measures φ 0 and φ 1 with compact support in R n which can be transported to each other at a finite W α cost, where W α (φ 0 , φ 1 ) := inf{M α (T ) : ∂T = φ 0 − φ 1 }, α ∈ [0, 1], the infimum is taken over real normal currents of finite mass and M α (T ) denotes the α-mass of T . Besides the class of α-irrigable measures (i.e., measures which can be transported to a Dirac measure with the appropriate total mass at a finite W α cost), two other important classes of measures are studied, which are called in the paper purely α-nonirrigable and marginally α-nonirrigable and are in a certain sense complementary to each other. For instance, purely α-nonirrigable and Ahlfors-regular measures are, roughly speaking, those having sufficiently high dimension. One shows that for φ 0 to be transported to φ 1 at finite W α cost their naturally defined purely α-nonirrigable parts have to coincide.