Resumen de Sharp inequalities for one-sided Muckenhoupt weights

Paul Alton Hagelstein , Ioannis Parissis , Olli Saari

• 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 \[w(\{x \in \mathbb{R} : M^{+}1_{E}(x) > \gamma\}) \leq c w(E)\] for all measurable sets E\subset \mathbb R. Furthermore, letting \[\mathcal{C}_{w}(\alpha) := \sup_{0 < w(E) < +\infty} \frac{1}{w(E)} w(\{x \in \mathbb{R} : M^{+}_{E}(x) > \alpha\})\] 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 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 \[p > e^{c[w]}_{A_{\infty}^{+}}\]