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Register a new userAsymptotically sharp reverse Holder inequalities for flat Muckenhoupt weights
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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 exponent in the reverse Holder inequality blows up as the weight becomes nearly constant. This is expressed in a precise and explicit computation of the constants involved in the reverse Holder inequality. The proofs avoid BMO methods and rely instead on precise covering arguments. Furthermore, in the one-dimensional case we prove sharp reverse Holder inequalities for one-sided and two sided weights in the sense that both the integrability exponent as well as the multiplicative constant appearing in the estimate are best possible. We also prove sharp endpoint weak-type reverse Holder inequalities and consider further extensions to general non-doubling measures and multiparameter weights.
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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 $p>e^{c[w]_{A_infty ^+}}$.
In this expository article we introduce a diagrammatic scheme to represent reverse classes of weights and some of their properties.
The Holder-Brascamp-Lieb inequalities are a collection of multilinear inequalities generalizing a convolution inequality of Young and the Loomis-Whitney inequalities. The full range of exponents was classified in Bennett et al. (2008). In a setting similar to that of Ivanisvili and Volberg (2015), we introduce a notion of size for these inequalities which generalizes $L^p$ norms. Under this new setup, we then determine necessary and sufficient conditions for a generalized Holder-Brascamp-Lieb type inequality to hold and establish sufficient conditions for extremizers to exist when the underlying linear maps match those of the convolution inequality of Young.
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 description of strongly symmetric homeomorphisms on the real line (a subclass of quasisymmetric homeomorphisms without using quasiconformal extensions.
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 inequalities through the ABP method, we construct a new convex coupling (i.e., a map that is the gradient of a convex function) between a generic set $E$ and the minimizer of the inequality (as in Gromovs proof of the isoperimetric inequality). Even if this map does not come from optimal transport, and even if there is a weight in the inequality, we adapt the methods of Figalli-Maggi-Pratelli and prove that if $E$ is almost optimal for the inequality then it is quantitatively close to a minimizer up to translations. Then, a delicate analysis is necessary to rule out the possibility of translations. As a step of our proof, we establish a sharp regularity result for restricted convex envelopes of a function that might be of independent interest.
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