Local integrability of strong and iterated maximal functions

Paul Alton Hagelstein. "Local integrability of strong and iterated maximal functions." Studia Mathematica 147.1 (2001): 37-50. <http://eudml.org/doc/284422>.

@article{PaulAltonHagelstein2001,
abstract = {Let $M_\{S\}$ 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 ℝ². A function h supported on the unit square Q = [0,1]×[0,1] is exhibited such that $∫_\{Q\} M_\{y\}M_\{x\}h < ∞$ but $∫_\{Q\} M_\{x\}M_\{y\}h = ∞$. It is shown that if f is a function supported on Q such that $∫_\{Q\} M_\{y\}M_\{x\}f < ∞$ but $∫_\{Q\} M_\{x\}M_\{y\}f = ∞$, then there exists a set A of finite measure in ℝ² such that $∫_\{A\} M_\{S\}f = ∞$.},
author = {Paul Alton Hagelstein},
journal = {Studia Mathematica},
keywords = {Hardy-Littlewood maximal function; strong maximal function; horizontal maximal function; vertical maximal function; Orlicz space},
language = {eng},
number = {1},
pages = {37-50},
title = {Local integrability of strong and iterated maximal functions},
url = {http://eudml.org/doc/284422},
volume = {147},
year = {2001},
}

TY - JOUR
AU - Paul Alton Hagelstein
TI - Local integrability of strong and iterated maximal functions
JO - Studia Mathematica
PY - 2001
VL - 147
IS - 1
SP - 37
EP - 50
AB - Let $M_{S}$ 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 ℝ². A function h supported on the unit square Q = [0,1]×[0,1] is exhibited such that $∫_{Q} M_{y}M_{x}h < ∞$ but $∫_{Q} M_{x}M_{y}h = ∞$. It is shown that if f is a function supported on Q such that $∫_{Q} M_{y}M_{x}f < ∞$ but $∫_{Q} M_{x}M_{y}f = ∞$, then there exists a set A of finite measure in ℝ² such that $∫_{A} M_{S}f = ∞$.
LA - eng
KW - Hardy-Littlewood maximal function; strong maximal function; horizontal maximal function; vertical maximal function; Orlicz space
UR - http://eudml.org/doc/284422
ER -