Sharp inequalities for one-sided Muckenhoupt weights

Abstract

Let $A_\infty ^+$ denote the class of one-sided Muckenhoupt weights, namely all the weights w for which $\mathsf {M}^+:L^p(w)\rightarrow L^{p,\infty }(w)$ 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

$$\begin{aligned} w(\{x \in \mathbb {R} : \, \mathsf {M}^+ \mathbf 1 _E (x)>\gamma \})\le cw(E) \end{aligned}$$

for all measurable sets $E\subset \mathbb R$. Furthermore, letting

$$\begin{aligned} \mathsf {C_w ^+}(\alpha ){:}{=}\sup _{0<w(E)<+\infty } \frac{1}{w(E)} w(\{x\in \mathbb R:\,\mathsf {M}^+ \mathbf 1 _E(x)>\alpha \}) \end{aligned}$$

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 Hölder inequality for one-sided Muckenhoupt weights, previously proved by Martín-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 $p>e^{c[w]_{A_\infty ^+}}$.