Journal of Mathematical Sciences, Vol. 157, No. 6, 2009 CONNECTING MEASURES BY MEANS OF BRANCHED TRANSPORTATION NETWORKS AT FINITE COST Emanuele Paolini Dipartimento di Matematica, Universit` a di Firenze P. S. Marco, 4, 50121 Firenze, Italy paolini@math.unifi.it Eugene Stepanov ∗ Dipartimento di Matematica, Universit` a di Pisa Largo B. Pontecorvo 5, 56127 Pisa, Italy e.stepanov@sns.it UDC 517.95 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. Bibliography: 19 titles. 1. Introduction If ϕ 0 and ϕ 1 are finite nonnegative Borel measures with compact support in R n , then for a fixed α ∈ [0, 1] one defines the α-cost of transporting measure ϕ 0 to the measure ϕ 1 by the formula W α (ϕ 0 ,ϕ 1 ) := inf {M α (T ): ∂T = ϕ 0 - ϕ 1 }, where the infimum is taken over all real flat chains of finite mass and M α (T ) denotes the α-mass of the flat chain T . Recently there is a lot of interest in the study of both W α costs and the optimal flat chains T transporting measure ϕ 0 to ϕ 1 and providing such a cost (i.e., providing the minimum in the above relationship). Such optimization problems arise quite naturally in ∗ To whom the correspondence should be addressed. Translated from Problems in Mathematical Analysis 39 February, 2009, pp. 65–79. 1072-3374/09/1576-0858 c  2009 Springer Science+Business Media, Inc. 858