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In a recent article J. Aldaz proved that the weak L1 bounds for the centeredmaximal operator associated to finite radial measures cannot be takenindependently with respect to the dimension. We show that at least for small pnear to 1 the same result holds for the Lp bounds of such measures withdecreasing densities. We also give some concrete examples, that includeGaussian measure, where better estimates with respect to the general case areobtained.

On the lack of dimension free estimates in Lp for maximal functions associated to radial measures