Growth of maximal functions

摘要

We consider the integrability of ϕ(f*) for variousmaximal functions f* and various increasing functions ϕ.We show that for some of the standard maximal functions arising in harmonicanalysis and ergodic theory, there is never integrability ofϕ(f*) for all Lebesgue integrable functions f exceptin cases where the growth of ϕ is slow enough so thatthe integrability follows from the standard weak maximalinequalities.