Sharp constants in the classical weak form of the John–Nirenberg inequality

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The sharp constants in the classical John–Nirenberg inequality are found by using Bellman function approach.

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Bellman function for extremal problems in BMO (2012, background): “It is widely known that for the John–Nirenberg inequality and for the reverse Hölder inequality for the Ap-weights, the Bellman functions can be constructed similarly (compare [12, 13, 9] and [14, 11, 7]).”