New York Journal of Mathematics
Volume 18 (2012) 523-549


S. Butler and J. Rosenblatt

Growth of maximal functions

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Published: July 9, 2012
Keywords: Maximal functions, maximal inequalities, Lebesgue derivatives, ergodic averages
Subject: 42B25, 28D05, 37A05

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


The second author was partially supported by NSF grant DMS 0555905

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Department of Mathematics, University of Illinois at Urbana-Champaign, 273 Altgeld Hall, 1409 West Green Street, Urbana, IL 61801, USA