During the talk I will discuss a result connected with
Alberti's rank-one theorem for gradients of functions of bounded
variations. Namely, I will present elements of our proof of a discrete
analog of this theorem for the setting of q-regular (Sobolev)
martingales. We derive it from a general description of polar
decomposition of vector measures on the boundary of the q-regular
infinite tree. This description in turns resembles De Philippis-Rindler
structure theorem. Joint work with D. Stolyarov and M. Wojciechowski.