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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.

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory
##### Keywords:
maximal functions; weighted inequalities; maximal operators
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