Geodesic
Some extremal problems for martingale transforms. I
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 51, Tome 527 (2023), pp. 5-53

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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. 
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     author = {V. I. Vasyunin and P. B. Zatitskii},
     title = {Some extremal problems for martingale transforms. {I}},
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V. I. Vasyunin; P. B. Zatitskii. Some extremal problems for martingale transforms. I. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 51, Tome 527 (2023), pp. 5-53. http://geodesic.mathdoc.fr/item/ZNSL_2023_527_a0/