Sharp inequalities for one-sided Muckenhoupt weights

Let $A_{\mathrm{\infty }}^{+}$ denote the class of one-sided Muckenhoupt weights, namely all the weights $w$ for which ${\mathsf{M}}^{+}:L^p(w)\to L^{p,\mathrm{\infty }}(w)$ for some $p>1$, where ${\mathsf{M}}^{+}$ is the forward Hardy-Littlewood maximal operator. We show that $w\in A_{\mathrm{\infty }}^{+}$ if and only if there exist numerical constants $\gamma \in (0,1)$ and $c>0$ such that $$w(\{x\in \mathbb{R}:{\mathsf{M}}^{+}{\mathbf{1}}_E(x)>\gamma \})\le cw(E)$$ for all measurable sets $E\subset \mathbb R$\mathrm{.}Furthermore,letting$$ \mathsf C_w ^+(\alpha):= \sup_{0<w(E)<+\infty} \frac{1}{w(E)} w({x\in\mathbb R:,\mathsf M^+\mathbf 1_E (x)>\alpha}) $$weshowthatforall$w\in A_\infty ^+$ we have the asymptotic estimate $\mathsf C_w ^+ (\alpha)-1\lesssim (1-\alpha)^\frac{1}{c[w]{A\infty ^+}}$for$\alpha$sufficientlycloseto$1$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 $A_{\mathrm{\infty }}^{+}$. Our methods also allow us to show that a weight $w\in A_{\mathrm{\infty }}^{+}$ satisfies $w\in A_p^{+}$ for all $p>e^{c[w{]}_{A_{\mathrm{\infty }}^{+}}}$.