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Maximal functions play a central role in the study of differentiation,

singular integrals and almost everywhere convergence. With Sergey Tikhonov

(ICREA, Barcelona) we recently proved some pointwise estimates for maximal

functions in terms of smoothness and rearrangements. I plan to discuss the recent

progress on these topics and some applications. In particular, I will discuss the

Fefferman-Stein inequality for the sharp maximal function for r.i. spaces which

are close to L∞.