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Functions with small BMO norm
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 08 January 2025, pp. 1-15
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- Article
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We characterize the functions with ‘small’ bounded mean oscillation (BMO) norm by establishing the precise connection between the space BMO and class
$A_\infty$ of Muckenhoupt weights. We prove that there exists a universal constant 
$c^*_2$ such that 
$\Vert f \Vert_{BMO} \lt c^*_2$ if and only if 
$\exp f \in A_2$, where 
$c^*_2$ is the sharp constant in the John and Nirenberg inequality. Similarly, in dimension one, we prove that 
$\Vert f \Vert_{BLO} \lt 1$ if and only if 
$\exp f \in A_1$. As application we introduce a structure of metric space in 
$A_\infty$ and prove that the closed unit ball of 
$A_\infty$ is a Banach space.
