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Register a new userTwo Equivalent Conditions for Muckenhoupt Weights
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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.
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We give a proof that every space of weighted square-integrable holomorphic functions admits an equivalent weight whose Bergman kernel has zeroes. Here the weights are equivalent in the sense that they determine the same space of holomorphic functions. Additionally, a family of radial weights in $L^1(mathbb{C})$ whose associated Bergman kernels have infinitely many zeroes is exhibited.
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 $p>e^{c[w]_{A_infty ^+}}$.
Bounded and compact differences of two composition operators acting from the weighted Bergman space $A^p_omega$ to the Lebesgue space $L^q_ u$, where $0<q<p<infty$ and $omega$ belongs to the class $mathcal{D}$ of radial weights satisfying a two-sided doubling condition, are characterized. On the way to the proofs a new description of $q$-Carleson measures for $A^p_omega$, with $p>q$ and $omegainmathcal{D}$, involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of $q$-Carleson measures for the classical weighted Bergman space $A^p_alpha$ with $-1<alpha<infty$ to the setting of doubling weights. The case $omegainwidehat{mathcal{D}}$ is also briefly discussed and an open problem concerning this case is posed.
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.
We extend the parameterization of sine-type functions in terms of conformal mappings onto slit domains given by Eremenko and Sodin to the more general case of generating functions of real complete interpolating sequences. It turns out that the cuts have to fulfill the discrete Muckenhoupt condition studied earlier by Lyubarskii and Seip.
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