# Sharp inequalities for one-sided Muckenhoupt weights

Research paper by Paul Hagelstein, Ioannis Parissis; Olli Saari

Indexed on: 31 Jul '17Published on: 17 Jun '17Published in: Collectanea Mathematica

#### Abstract

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 $$p>e^{c[w]_{A_\infty ^+}}$$ .AbstractLet $$A_\infty ^+$$ 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 $$p>e^{c[w]_{A_\infty ^+}}$$ . $$A_\infty ^+$$ $$A_\infty ^+$$w $$\mathsf {M}^+:L^p(w)\rightarrow L^{p,\infty }(w)$$ $$\mathsf {M}^+:L^p(w)\rightarrow L^{p,\infty }(w)$$ $$p>1$$ $$p>1$$ $$\mathsf {M}^+$$ $$\mathsf {M}^+$$ $$w\in A_\infty ^+$$ $$w\in A_\infty ^+$$ $$\gamma \in (0,1)$$ $$\gamma \in (0,1)$$ $$c>0$$ $$c>0$$ \begin{aligned} w(\{x \in \mathbb {R} : \, \mathsf {M}^+ \mathbf 1 _E (x)>\gamma \})\le cw(E) \end{aligned} \begin{aligned} w(\{x \in \mathbb {R} : \, \mathsf {M}^+ \mathbf 1 _E (x)>\gamma \})\le cw(E) \end{aligned} $$E\subset \mathbb R$$ $$E\subset \mathbb R$$ \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} \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} $$w\in A_\infty ^+$$ $$w\in A_\infty ^+$$ $$\mathsf {C_w ^+}(\alpha )-1\lesssim (1-\alpha )^\frac{1}{c[w]_{A_\infty ^+}}$$ $$\mathsf {C_w ^+}(\alpha )-1\lesssim (1-\alpha )^\frac{1}{c[w]_{A_\infty ^+}}$$ $$\alpha$$ $$\alpha$$ $$c>0$$ $$c>0$$ $$A_\infty ^+$$ $$A_\infty ^+$$ $$w\in A_\infty ^+$$ $$w\in A_\infty ^+$$ $$w\in A_p ^+$$ $$w\in A_p ^+$$ $$p>e^{c[w]_{A_\infty ^+}}$$ $$p>e^{c[w]_{A_\infty ^+}}$$