On the \(A_\infty\) conditions for general bases (Q269895)

It is known that the usual Muckenhoupt weight class \(A_\infty\), defined via balls or cubes, has several different but equivalent descriptions, such as the union of \(A_p\) with \(p\geq1\), or the limit of \(A_p\) as \(p\to \infty\), or an exponential inequality. However, these equivalences may be not true for \(A_\infty\) associated with a general basis. Recall that a basis \(\mathcal{B}\) in a measure space \((X,\mu)\) is a collection of \(\mu\)-measurable subsets \(B\) of \(X\) with \(0<\mu(B)<\infty\). Via replacing balls or cubes in the definitions of the usual Muckenhoupt class \(A_\infty\) by sets from \(\mathcal{B}\), one gets the corresponding general \(A_\infty\) class with respect to \(\mathcal{B}\). It is known that, if \(X=(0,\infty)\) and \(\mathcal{B}\) consists of all intervals of form \((0,b)\), then the union of the related \(A_p\) classes is strictly contained in the \(A_\infty\) class given by exponential inequalities. This paper is devoted to discussing and comparing different characterizations of \(A_\infty\) classes with respect to a general basis \(\mathcal{B}\), and some equivalence, implication and restricted implication relations among these characterizations are obtained. For some special basis, the authors also get un-equivalence results. Finally, the authors define BMO spaces associated with \(\mathcal{B}\) and show that the classical result relating functions in BMO to logarithms of \(A_\infty\) weights does not hold in general.