The maximal theorem for weighted grand Lebesgue spaces

Alberto Fiorenza, Babita Gupta, and Pankaj Jain. "The maximal theorem for weighted grand Lebesgue spaces." Studia Mathematica 188.2 (2008): 123-133. <http://eudml.org/doc/285100>.

@article{AlbertoFiorenza2008,
abstract = {We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0,1) ⊂ ℝ, and the maximal function is localized in (0,1). Moreover, we prove that the inequality $||Mf||_\{p),w\} ≤ c||f||_\{p),w\}$ holds with some c independent of f iff w belongs to the well known Muckenhoupt class $A_\{p\}$, and therefore iff $||Mf||_\{p,w\} ≤ c||f||_\{p,w\}$ for some c independent of f. Some results of similar type are discussed for the case of small Lebesgue spaces.},
author = {Alberto Fiorenza, Babita Gupta, Pankaj Jain},
journal = {Studia Mathematica},
keywords = {maximal function; Muckenhoupt weights; grand Lebesgue spaces; decreasing rearrangements; Hardy inequality},
language = {eng},
number = {2},
pages = {123-133},
title = {The maximal theorem for weighted grand Lebesgue spaces},
url = {http://eudml.org/doc/285100},
volume = {188},
year = {2008},
}

TY - JOUR
AU - Alberto Fiorenza
AU - Babita Gupta
AU - Pankaj Jain
TI - The maximal theorem for weighted grand Lebesgue spaces
JO - Studia Mathematica
PY - 2008
VL - 188
IS - 2
SP - 123
EP - 133
AB - We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0,1) ⊂ ℝ, and the maximal function is localized in (0,1). Moreover, we prove that the inequality $||Mf||_{p),w} ≤ c||f||_{p),w}$ holds with some c independent of f iff w belongs to the well known Muckenhoupt class $A_{p}$, and therefore iff $||Mf||_{p,w} ≤ c||f||_{p,w}$ for some c independent of f. Some results of similar type are discussed for the case of small Lebesgue spaces.
LA - eng
KW - maximal function; Muckenhoupt weights; grand Lebesgue spaces; decreasing rearrangements; Hardy inequality
UR - http://eudml.org/doc/285100
ER -