Vol. 112, No. 2, 1984

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On the reverse weak type inequality for the Hardy maximal function and the weighted classes L(log L)k

Kenneth F. Andersen and Wo-Sang Young

Vol. 112 (1984), No. 2, 257–264

Muckenhoupt has given a necessary and sufficient condition to be satisfied by the weight functions U and V in order that the Hardy-Littlewood maximal function Mf should satisfy a weighted weak type (1,1) inequality. In this note, conditions on the weight functions U and V are given in order that the sense of this inequality may be reversed. This is then applied to give conditions which ensure that the integrability of Mf with respect to a weight implies that f belongs to a weighted Zygmund class Llog L, thus extending a result of Stein. Analogous results related to the strong maximal function and the classes L(log L)k are also given. These extend certain results of Favo, Gatto and Gutiérrez.

Mathematical Subject Classification 2000
Primary: 42B25
Received: 11 June 1982
Published: 1 June 1984
Kenneth F. Andersen
Wo-Sang Young