Local integrability of strong and iterated maximal functions

Paul Alton Hagelstein¹

¹ Department of Mathematics Princeton University Princeton, NJ 08544, U.S.A.

Studia Mathematica, Tome 147 (2001) no. 1, pp. 37-50

Résumé

Let $M_{\rm S}$ denote the strong maximal operator. Let $M_{x}$ and $M_{y}$ denote the one-dimensional Hardy–Littlewood maximal operators in the horizontal and vertical directions in ${\mathbb R}^{2}$. A function $h$ supported on the unit square $Q = [0,1] \times [0,1]$ is exhibited such that $\int _{Q}M_{y}M_{x}h \infty $ but $\int _{Q}M_{x}M_{y}h = \infty $. It is shown that if $f$ is a function supported on $Q$ such that $\int _{Q}M_{y}M_{x}f \infty $ but $\int _{Q}M_{x}M_{y}f = \infty $, then there exists a set $A$ of finite measure in ${\mathbb R}^{2}$ such that $\int _{A}M_{\rm S}f = \infty $.

Détail

DOI : 10.4064/sm147-1-4

Keywords: denote strong maximal operator denote one dimensional hardy littlewood maximal operators horizontal vertical directions mathbb function supported unit square times exhibited int infty int infty shown function supported int infty int infty there exists set finite measure mathbb int infty