Sharp inequalities for one-sided Muckenhoupt weights

Let $A_\infty ^+$ denote the class of one-sided Muckenhoupt weights, namely all the weights $w$ for which $\mathsf M^+:L^p(w)\to L^{p,\infty}(w)$ for some $p>1$, where $\mathsf M^+$ is the forward Hardy-Littlewood maximal operator. We show that $w\in A_\infty ^+$ if and only if there exist numerical constants $\gamma\in(0,1)$ and $c>0$ such that $$w(\{x \in \mathbb{R} : \, \mathsf M ^+\mathbf 1_E (x)>\gamma\})\leq c w(E)$$ for all measurable sets $E\subset \mathbb R$. Furthermore, letting $$\mathsf C_w ^+(\alpha):= \sup_{0<w(E)<+\infty} \frac{1}{w(E)} w(\{x\in\mathbb R:\,\mathsf M^+\mathbf 1_E (x)>\alpha\})$$ we show that for all $w\in A_\infty ^+$ we have the asymptotic estimate $\mathsf C_w ^+ (\alpha)-1\lesssim (1-\alpha)^\frac{1}{c[w]_{A_\infty ^+}}$ for $\alpha$ sufficiently close to $1$ and $c>0$ a numerical constant, and that this estimate is best possible. We also show that the reverse H\"older inequality for one-sided Muckenhoupt weights, previously proved by Mart\'in-Reyes and de la Torre, is sharp, thus providing a quantitative equivalent definition of $A_\infty ^+$. Our methods also allow us to show that a weight $w\in A_\infty ^+$ satisfies $w\in A_p ^+$ for all $p>e^{c[w]_{A_\infty ^+}}$.