\(A_ \infty\) type conditions for general measures in \(R^ 1\) (Q1194673)

The authors give results for \(A_ \infty\) with respect to Borel measures finite on compact subsets of \(R\). Most prior results had assumed that such measures satisfied a doubling condition. The authors prove some of their results by using results about one-sided maximal functions. For the Hardy-Littlewood maximal operator on \(R^ n\), \[ Mf(x)=\sup_{x\in Q} {1\over| Q|} \int_ Q | f(y)| dy, \] the \(A_ p\) class controls weighted estimates. We say that a function \(w\in A_ p\), \(1<p<\infty\), if \[ A_ p(w)=\sup_ Q \left({1\over| Q|} \int_ Q w\right)^{1/p} \left({1\over | Q|} \int_ Q w^{1-p'}\right)^{1/p'} < +\infty. \] One has \[ \int Mf(x)^ p w\leq C^ p \int | f(x)|^ p w\Longleftrightarrow w\in A_ p. \] When \(\mu\) is a nonnegative Borel measure finite on compact sets, the corresponding maximal operator is \[ M_ \mu f(x)= \sup_{x\in Q} {1\over\mu(Q)} \int_ Q | f| d\mu \] and \(w\in A_ p(\mu)\) iff \[ A_{p,\mu}(w)=\sup_ Q \left({1\over\mu(Q)} \int_ Q w d\mu\right)^{1/p} \left({1\over \mu(Q)} \int_ Q w^{1-p'} d\mu\right)^{1/p'} <+\infty. \] The one-sided maximal function on \(R\) is \[ M^ +_ \mu f(x)=\sup_{h>0} {1\over \mu[x,x+h)}\int_{[x,x+h)} | f(y)| d\mu, \] or \[ M^ -_ \mu f(x)=\sup_{h>0} {1\over \mu(x-h,x]}\int_{(x-h,x]} | f(y)| d\mu, \] and \(w\in A^ +_ p(\mu)\) if \[ \left(\int^ b_ a w d\mu\right)^{1/p}\left(\int^ c_ b w^{1-p'} d\mu\right)^{1/p'}\leq K\mu(a,c) \] with some \(K\) independent of \(a<b<c\), and similarly, \(w\in A^ -_ p\) if \[ \left(\int^ c_ b w d\mu\right)^{1/p}\left(\int^ b_ a w^{1- p'} d\mu\right)^{1/p'}\leq K\mu(a,c). \] \textit{K. F. Andersen} [Trans. Am. Math. Soc. 326, No. 2, 907-920 (1991; Zbl 0736.42013)] showed how to extend the \(A_ p\) and \(A^ +_ p\) theory to generalized measures on \(R\). The authors obtain the basic results in such a situation for \(A_ \infty\) and \(A^ +_ \infty\). They say that \(\nu\in A^ +_ \infty(\mu)\) if there exists \(K,\delta>0\) such that for all \(a< b< c\) and \(E\subseteq [b,c)\), \({\mu(E)\over \mu(a,c)}\leq K\Bigl({\nu(E)\over \nu(a,b]}\Bigr)^ \delta\). The fact that \(\nu\in A^ +_ \infty\) implies \(\mu\ll\nu\) but not that \(\nu\ll\mu\). This latter fact is assumed because they want to study the relationship between \(\nu\in A^ +_ \infty(\mu)\) and \(\mu\in A^ -_ \infty(\nu)\). We will write \(d\nu=w d\mu\). Their main result is a set of equivalent conditions characterizing \(A^ +_ \infty\) including the facts that \(A^ +_ \infty=\bigcup_ p A^ +_ p\), that \(1/w\in A^ -_ \infty\), and a Hruščev type condition: \(\exists K\) such that for all \(a< b< c\) satisfying \(\mu(a,b)\leq{1\over 2}\mu(a,c)\leq \mu(a,b]\), \[ {\nu(a,b]\over \mu(a,b]}\exp\left({1\over \mu[b,c)} \int_{[b,c)} \log 1/w d\mu\right) \leq K. \] They use this result to prove the major facts about their general \(A_ \infty(\mu)\) (recall that \(\mu\) need not be doubling) including a Hruščev type characterization, a reverse Hölder characterization and a Fujii characterization: \(\nu\in A_ \infty(\mu)\) iff there exists a constant \(K\) such that \[ \sup_ I {1\over\nu(I)} \int_ I \log^ +(w(x)\mu(I)/\nu(I))w(x)d\mu(x)\leq K. \] {}.