Behavior of maximal functios inRn for largen
What is noteworthy here is that any of the usual covering arguments lead only to a weak-type (1,1) bound which grows exponentially in n, and thus by interpolation one obtains by this method (1.1) with Ap replaced by a bound which increases exponentially in n. Thus the foUowing further questions now present themselves: (1) Does M (") have a weak-type (1, 1) bound independent of n? (2) What can be said when the usual balls are replaced by dilates of more general sets?, We give here some partial answers to these questions: