Some extremal problems for martingale transforms. I

With this paper, we begin a series of studies of extremal problems for estimating the distributions of martingale transforms of bounded martingales. The Bellman functions corresponding to such problems are pointwise minimal diagonally concave functions on a horizontal strip, satisfying certain given boundary conditions. We describe the basic structures that arise in the construction such functions and present a solution in the case \break of asymmetric boundary conditions and a sufficiently small width of the strip.