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Sharp inequalities for one-sided Muckenhoupt weights (1601.00938v1)

Published 5 Jan 2016 in math.CA

Abstract: Let $A_\infty +$ denote the class of one-sided Muckenhoupt weights, namely all the weights $w$ for which $\mathsf M+:Lp(w)\to L{p,\infty}(w)$ 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 $p>e{c[w]{A\infty +}}$.

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