Another characterization of the two-weight norm inequalities for the maximal operators (Q1112281)

For a Borel measure \(\nu\) on \({\mathbb{R}}^ n\), and \(0<\alpha \leq 1\), set \(M_{\alpha}\nu (x)=\sup (1/| Q|^{\alpha})\int_{Q}| d\nu |,\) where the supremum is taken over all the cubes Q in \({\mathbb{R}}^ n\) which contans x and \(| Q|\) denotes the Lebesgue measure of Q. E. T. Sawyer showed that for a nonnegative measure \(\omega\) and a nonnegative function v(x) on \({\mathbb{R}}^ n\), and for \(0<\alpha \leq 1\), \(1<p\leq q<\infty\), there exists \(C_ 1>0\) such that \((\int [M_{\alpha}f]^ q d\omega)^{1/q}\leq C_ 1(\int | f|^ pv dx)^{1/p},\) if and only if there exists \(C_ 2>0\) such that \(\int_{Q}[M_{\alpha}(\chi_ Qv^{1-p'})]^ q d\omega \leq C_ 2(\int_{Q}v^{1-p'} dx)^{q/p},\) where \((1-p)(1-p')=1\) and \(\chi_ Q\) denotes the characteristic function of Q. The author, returning to an original idea of B. Muckenhoupt, gives the following theorem, from which Sawyer's theorem can be deduced. Theorem. Let \(\omega\), \(\sigma\) be nonnegative Borel measures on \({\mathbb{R}}^ n\). Suppose \(1\leq q<\infty\), \(0<\alpha \leq 1\), \(0<\delta <1\), and fix a cube \(Q\subset {\mathbb{R}}^ n\). Suppose there exist a nonnegative Borel measure \(\mu_ Q\) and \(C_ Q>0\) such that \(\int_{E}d\omega ((1/| I|^{\alpha})\int_{I}d\sigma)^ q\leq C_ Q\int_{E}d\mu_ Q\) for any subcube I of Q and any measurable subset E of I with \(| E| \geq \delta | I|\). Then there exists \(c_ 0>0\) such that \(\int_{Q}[M_{\alpha}(\chi_ Q\sigma)(s)]^ q d\omega \leq c_ 0Q_ Q\int_{Q}d\mu_ Q.\) Some other consequences are given.