Sharp constants for John-Nirenberg and Reverse Hölder inequalities are the same on the circle, the interval, and the line, proved via a new martingale-based transference principle.
The John--Nirenberg constant of ${\rm BMO}^p,$ $1\le p\le 2$
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abstract
We compute the exact John--Nirenberg constant of ${\rm BMO}^p((0,1))$ for $1\le p\le 2,$ which has been known only for $p=1$ and $p=2.$ We also show that this constant is attained in the weak-type John--Nirenberg inequality and obtain a sharp lower estimate for the distance in ${\rm BMO}^p$ to $L^\infty.$ These results rely on sharp $L^p$- and weak-type estimates for logarithms of $A_\infty$ weights, which in turn use the exact expressions for the corresponding Bellman functions.
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math.CA 1years
2019 1verdicts
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Sharp transference principle for $\mathrm{BMO}$ and $A_p$
Sharp constants for John-Nirenberg and Reverse Hölder inequalities are the same on the circle, the interval, and the line, proved via a new martingale-based transference principle.