# Sharp inequalities for one-sided Muckenhoupt weights

Research paper by **Paul A. Hagelstein, Ioannis Parissis, Olli Saari**

Indexed on: **05 Jan '16**Published on: **05 Jan '16**Published in: **Mathematics - Classical Analysis and ODEs**

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#### Abstract

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
$p>e^{c[w]_{A_\infty ^+}}$.