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28 March 2024
 
  » arxiv » 1501.0097

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Inequalities for BMO on $alpha$-trees
Leonid Slavin ; Vasily Vasyunin ;
Date 31 Dec 2014
AbstractWe develop technical tools that enable the use of Bellman functions for BMO defined on $alpha$-trees, which are structures that generalize dyadic lattices. As applications, we prove the integral John--Nirenberg inequality and an inequality relating $L^1$- and $L^2$-oscillations for BMO on $alpha$-trees, with explicit constants. When the tree in question is the collection of all dyadic cubes in $mathbb{R}^n,$ the inequalities proved are sharp. We also reformulate the John--Nirenberg inequality for the continuous BMO in terms of special martingales generated by BMO functions. The tools presented can be used for any function class that corresponds to a non-convex Bellman domain.
Source arXiv, 1501.0097
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