Sharp results in the integral-form John–Nirenberg inequality

Slavin, L.; Vasyunin, V.

doi:10.1090/S0002-9947-2011-05112-3

Sharp results in the integral-form John–Nirenberg inequality
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by L. Slavin and V. Vasyunin PDF

Trans. Amer. Math. Soc. 363 (2011), 4135-4169 Request permission

Abstract:

We consider the strong form of the John-Nirenberg inequality for the $L^2\!$-based BMO. We construct explicit Bellman functions for the inequality in the continuous and dyadic settings and obtain the sharp constant, as well as the precise bound on the inequality’s range of validity, both previously unknown. The results for the two cases are substantially different. The paper not only gives another instance in the short list of such explicit calculations, but also presents the Bellman function method as a sequence of clear steps, adaptable to a wide variety of applications.

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