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A remark on weighted inequalities for general maximal operators. (English) Zbl 0810.42008
For a basis \(\mathcal B\) on \(\mathbb{R}^ n\), define the maximal function \(M_{{\mathcal B}}\) by \[ M_{{\mathcal B}} f(x)= \sup_{x\in B\in {\mathcal B}} {1\over | B|}\int_ B | f(y)| dy. \] The author studies the boundedness of these maximal operators from \(L^ p_ v\) to \(L^ p_ w\). The main result proved is that \(M_{{\mathcal B}}\) defines a bounded operator from \(L^ p_ v\) to \(L^ p_ w\) if the following conditions are satisfied: 1. \(M_{{\mathcal B}}: L^ p\to L^ p\) for \(1< p<\infty\); 2. there exists constants \(\lambda\) and \(c\), \(0< \lambda< 1\) and \(c>0\), so that for all measurable sets \(E\subset \mathbb{R}^ n\), \[ w(\{x\in \mathbb{R}^ n: M_{{\mathcal B}}(\chi_ E)(x)> \lambda\})\leq cw(E); \] 3. there are constants \(r,c> 1\) so that for all \(B\in {\mathcal B}\), \[ \left({1\over | B|}\int_ B w(y)dy\right)\left({1\over | B|} \int_ B v(y)^{(1- p')r}dy\right)^{{p-1\over r}}\leq c. \] This result generalizes work of Sawyer, Jawerth, and Neugebauer.

42B25 Maximal functions, Littlewood-Paley theory
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