Homeomorphisms preserving \(A_ p\)
A\({}_ p\) denotes the set of weights w satisfying the condition \[ A_ p(w)=\sup_{Q}(| Q|^{-1}\int_{Q}wdx)(| Q|^{- 1}\int_{Q}w^{1-p'}dx)^{p-1}<\infty,\quad 1<p<\infty,\quad 1/p+1/p'=1. \] \textit{B. Muckenhoupt} [Trans. Am. Math. Soc. 165, 207-226 (1972; Zbl 0236.26016)] showed that for \(w\in A_ p\), it holds \(\int Mf(x)^ pw(x)dx\leq C^ p\int f(x)^ pw(x)dx,\) where Mf(x) is the Hardy-Littlewood maximal function of f(x)\(\geq 0\). The similar inequality in which the maximal function is replaced by the Hilbert transform, holds. It is known that \(w(1/x)\in A_ 2\) for any \(w(x)\in A_ 2\). The paper discusses the more general question: which homeomorphisms of \({\mathbb{R}}\) preserve the \(A_ p\) class? Assume that h: \({\mathbb{R}}\to {\mathbb{R}}\) is a homeomorphism onto such that \(h,h^{-1}\) are locally absolute continuous and \(h'\geq 0\). Use the notation \(Q\prec P\) to mean that Q is majorized by and expression depending on P only so that for \(P\leq T\), \(Q\prec T.One\) of the main theorems of the paper is: Theorem. Let h: \({\mathbb{R}}\to {\mathbb{R}}\) be a homeomorphism as above, and suppose \(1<p_ 0<\infty\), \(0\leq \alpha \leq 1\). Then, for every \(w\in A_{p_ 0}\), \(w\circ h\cdot h^{'\alpha}\in A_{p_ 0}\) with \(A_{p_ 0}(w\circ h\cdot h^{'\alpha})\prec A_{p_ 0}(w)\) if and only if \(h'\in \cap_{p>1}A_ p.\) The paper investigates to determine the pointwise multipliers of \(A_ p\), the corresponding questions for \(A_{\infty}\) by means of a precise connection between reverse Hölder inequalities and \(A_{\infty}\), and also for double weights \((u,v)\in A_ p\).
scientific article; zbMATH DE number 4109327