On the maximal operator associated with certain rotational invariant measures

The aim of this work is to investigate the integrability properties of the maximal operator M µ, associated with a non-doubling measure µ defined on .n. We start by establishing for a wide class of radial and increasing measures µ, that M µ is bounded on all the spaces L pµ (.n), p > 1. Also, we show that there is a radial and increasing measure µ for which M µ does not map L pµ (.n) into weak L pµ (.n), 1 . p < ..