, Volume 67, Issue 1, pp 95-108

Weighted weak type integral inequalities for the Hardy-Littlewood maximal operator

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Abstract

In this paper we characterize the pairs of weights (u, w) for which the Hardy-Littlewood maximal operatorM satisfies a weak type integral inequality of the form $$\smallint _{\left\{ {x \in R^n :Mf(x) > \lambda } \right\}} udx \leqq \frac{C}{{\phi (\lambda )}}\smallint _{R^n } \phi (|f|)wdx$$ withC independent off andλ>0, whereφ is anN-function. Moreover, for a given weightw, a necessary and sufficient condition is found for the existence of a positive weightu such thatM satisfies an integral inequality as above. Lastly, in the caseu=w, we notice that the conclusion of the extrapolation theorem given by J. L. Rubio de Francia, which appeared in Am. J. Math.106 (1984), can be strengthened to Orlicz spaces.