 
Summary: ON CERTAIN HARDYLITTLEWOOD TYPE MAXIMAL
OPERATORS
PARTHENA AVRAMIDOU
Abstract. In this expository paper we study the boundedness of certain
HardyLittlewood type maximal operators on Lp spaces in real variable har
monic analysis and in ergodic theory, underlining the interplay between the
two areas.
1. Introduction
Our discussion begins with the celebrated Differentiation Theorem of Lebesgue (
on R in [27] and on Rn
in [26]) and the Local Ergodic Theorem of Wiener [46]. The
formal resemblance only hints towards the deep connection between real variable
theory and ergodic theory. Many results in one area suggest analogues in the other,
and there is an exchange of techniques that goes both ways. One of the goals of
this survey article is to comment on some instances of this phenomenon.
Theorem 1.1 (Lebesgue Differentiation Theorem, 1904). If f is locally integrable
on R, then
lim
0+
1
