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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 $win A_infty ^+$ if and only if there exist numerical constants $gammain(0,1)$ 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 $Esubset mathbb R$. Furthermore, letting $$ mathsf C_w ^+(alpha):= sup_{0<w(E)<+infty} frac{1}{w(E)} w({xinmathbb R:,mathsf M^+mathbf 1_E (x)>alpha}) $$ we show that for all $win A_infty ^+$ we have the asymptotic estimate $mathsf C_w ^+ (alpha)-1lesssim (1-alpha)^frac{1}{c[w]_{A_infty ^+}}$ 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 Holder inequality for one-sided Muckenhoupt weights, previously proved by Martin-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 $win A_infty ^+$ satisfies $win A_p ^+$ for all $p>e^{c[w]_{A_infty ^+}}$.
We present reverse Holder inequalities for Muckenhoupt weights in $mathbb{R}^n$ with an asymptotically sharp behavior for flat weights, namely $A_infty$ weights with Fujii-Wilson constant $(w)_{A_infty}to 1^+$. That is, the local integrability expone
We present a new characterization of Muckenhoupt $A_{infty}$-weights whose logarithm is in $mathrm{VMO}(mathbb{R})$ in terms of vanishing Carleson measures on $mathbb{R}_+^2$ and vanishing doubling weights on $mathbb{R}$. This also gives a novel desc
We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homogeneous weights. Inspired by the proof of such isoperimetric inequali
Let $ Tf =sum_{ I} varepsilon_I langle f,h_{I^+}rangle h_{I^-}$. Here, $ lvert varepsilon _Irvert=1 $, and $ h_J$ is the Haar function defined on dyadic interval $ J$. We show that, for instance, begin{equation*} lVert T rVert _{L ^{2} (w) to L ^{2}
The purpose of this paper is to establish some one-sided estimates for oscillatory singular integrals. The boundedness of certain oscillatory singular integral on weighted Hardy spaces $H^{1}_{+}(w)$ is proved. It is here also show that the $H^{1}_{+