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 ${\mathsf{M}}^{+}:L^p(w)\to L^{p,\infty }(w)$ 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 $$w(\{x\in \mathbb{R}:{\mathsf{M}}^{+}{1}_E(x)>\gamma \})\le cw(E)$$ for all measurable sets $E\subset \mathbb{R}$. Furthermore, letting $${\mathsf{C}}_w^{+}(\alpha ):=\underset{0<w(E)<+\infty }{\sup }\frac{1}{w(E)}w(\{x\in \mathbb{R}:{\mathsf{M}}^{+}{1}_E(x)>\alpha \})$$ 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 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 $p>e^{c[w{]}_{A_{\infty }^{+}}}$.

11 pages, submitted for publication

References in corpus (3)

• Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases
• Parabolic BMO and global integrability of supersolutions to doubly nonlinear parabolic equations
• Weighted Solyanik Estimates for the Hardy-Littlewood maximal operator and embedding of $A_{\infty }$ into $A_p$