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

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 % MathType!MTEF!2!1!+-% feaafiart1ev1aaatuuDJXwAK1uy0Hwmaerbfv3ySLgzG0uy0Hgip5% wzamXvP5wqonvsaeHbfv3ySLgzaeXatLxBI9gBamXvP5wqSXMqHnxA% Jn0BKvguHDwzZbqehqvATv2CG4uz3bIuV1wyUbqehm0B1jxALjhiov% 2DaebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qq% aqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8% qqQ8frFve9Fve9Ff0dmeaaciGacmaadaWabiqaeaqbaqaagaaakeaa% imaajqgaGdGamaO5-TIiYRWaaSbaaSqaamaacmaabaaceiGaa4hEai% abgIGioJabaiaa9jfadaahaaadbeqaaiaa+5gaaaWccaqF6aGaa4xt% aiaa+zgacaqFOaGaa4hEaiaa9LcacaqF+aGaeq4UdWgacaGL7bGaay% zFaaaabeaakiaa+vhacaGFKbGaa4hEaeXbbjxAHXgaiCaacaaFGaWe% fv3ySLgznfgDOjdarGqr1ngBPrginfgDObcv39gaiGqacqWEMjIHda% Wcaaqaaiaa+neaaeaaimGacqGEgpGzcaqFOaGaeO3UdWMaa0xkaaaa% jqgaGdGamaO5-TIiYRWaaSbaaSqaaiaa9jfadaahaaadbeqaaiaa+5% gaaaaaleqaaOGaeONXdyMaa0hkaiaa9XhacGGva6NzaiaccayF8bGa% iiaG9LcacGGaa63DaiaccaOFKbGaiiaG+Hhaaaa!8017! $$\smallint _{\left\{ {x \in R^n :Mf(x) > \lambda } \right\}} udx \leqq \frac{C}{{\phi (\lambda )}}\smallint _{R^n } \phi (|f|)wdx$$ 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.