Related papers: Sharp estimates involving $A_\infty$ and $LlogL$ c…

In a recent paper V. Vasyunin presented a proof of the reverse H\"older inequality with sharp constants for the weights satisfying the usual Muckenhoupt condition. In this paper we present the inverse, that is, we use the Bellman function…

In this article we present a new proof of a sharp Reverse H\"older Inequality for $A_\infty$ weights that is valid in the context of spaces of homogeneous type. Then we derive two applications: a precise open property of Muckenhoupt classes…

In the dyadic case the union of the Reverse H\"{o}lder classes, $RH_p^d$ is strictly larger than the union of the Muckenhoupt classes $ A_p^d$. We introduce the $RH_1^d$ condition as a limiting case of the $RH_p^d$ inequalities as $p$ tends…

In this expository article we collect and discuss some recent results on different consequences of a Sharp Reverse H\"older Inequality for $A_{\infty}$ weights. For two given operators $T$ and $S$, we study $L^p(w)$ bounds of…

We get sharp estimates for the distribution function of nonnegative weights, which satisfy so called $A_{p_1, p_2}$ condition. For particular choices of parameters $p_1$, $p_2$ this condition becomes an $A_p$-condition or Reverse H\"{o}lder…

We provide new quantitative results on the embedding of the Muckenhoupt class $A_\infty$ into $A_p$ with the correct asymptotic behavior when the Fujii--Wilson constant $[w]_{A_\infty}$ is close to 1, namely that the parameter $p$ goes to 1…

We present reverse H\"older 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…

We prove an appropriate sharp quantitative reverse H\"older inequality for the $C_p$ class of weights from which we obtain as a limiting case the sharp reverse H\"older inequality for the $A_\infty$ class of weights. We use this result to…

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…

As a corollary to our main result we deduce sharp A_p$ inequalities for T being either the Hilbert transform in dimension d=1, the Beurling transform in dimension d=2, or a Riesz transform in any dimension d\ge 2. For T_{\ast} the maximal…

In this article, we establish a nearly sharp localized weighted inequality related to Gagliardo and Sobolev seminorms, respectively, with the sharp $A_1$-weight constant or with the specific $A_p$-weight constant when $p\in (1,\infty)$. As…

We study the $P_1$ finite element approximation of the best constant in the classical Hardy inequality over bounded domains containing the origin in $\mathbb{R}^N$, for $N \geq 3$. Despite the fact that this constant is not attained in the…

We give a new proof of the sharp one weight $L^p$ inequality for any operator $T$ that can be approximated by Haar shift operators such as the Hilbert transform, any Riesz transform, the Beurling-Ahlfors operator. Our proof avoids the…

The purpose of this paper is to prove pointwise inequalities and to establish the boundedness on weighted $L^{p}$ spaces for pseudo-differential operators $T_{a}$ defined by the symbol $a\in S^{m}_{\varrho,\delta}$ with $0\leq\varrho\leq1,$…

Global weighted $L^{p}$-estimates are obtained for the gradient of solutions to a class of linear singular, degenerate elliptic Dirichlet boundary value problems over a bounded non-smooth domain. The coefficient matrix is symmetric,…

Muckenhoupt and Reverse H\"{o}lder classes of weights play an important role in harmonic analysis, PDE's and quasiconformal mappings. In 1974 Coifman and Fefferman showed that a weight belongs to a Muckenhoupt class $A_p$ for some…

We give a simpler proof of the a priori estimates obtained in the paper by Duran, Sanmartino and Toschi for solutions of elliptic problems in weighted Sobolev norms when the weights belong to the Muckenhoupt class $A_p$. The argument is a…

We give sharp remainder terms of $L^{p}$ and weighted Hardy and Rellich inequalities on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. As consequences, we obtain analogues of the generalised…

In this article, with introducing concepts of variable scalar $\mathcal{A}_{p(\cdot),\infty}$ weights and variable matrix $\mathscr{A}_{p(\cdot),\infty}$ weights, we seek a comprehensive theory of $A_\infty$ weights within the framework of…

We study a class of non-divergence form elliptic and parabolic equations with singular first-order coefficients in an upper half space with the homogeneous Dirichlet boundary condition. In the simplest setting, the operators in the…