On a converse inequality for maximal functions in Orlicz spaces

Kita, H.. "On a converse inequality for maximal functions in Orlicz spaces." Studia Mathematica 118.1 (1996): 1-10. <http://eudml.org/doc/216260>.

@article{Kita1996,
abstract = {Let $Φ(t) = ʃ_\{0\}^\{t\} a(s)ds$ and $Ψ(t) = ʃ_\{0\}^\{t\} b(s)ds$, where a(s) is a positive continuous function such that $ʃ_\{1\}^\{∞\} a(s)/s ds = ∞$ and b(s) is quasi-increasing and $lim_\{s→∞\}b(s) = ∞$. Then the following statements for the Hardy-Littlewood maximal function Mf(x) are equivalent: (j) there exist positive constants $c_1$ and $s_\{0\}$ such that $ʃ_\{1\}^\{s\} a(t)/t dt ≥ c_\{1\}b(c_\{1\}s)$ for all $s ≥ s_\{0\}$; (jj) there exist positive constants $c_2$ and $c_3$ such that $ʃ_\{0\}^\{2π\} Ψ((c_2)/(|⨍|_\{\}) |⨍(x)|) dx ≤ c_3 + c_\{3\} ʃ_\{0\}^\{2π\} Φ(1/(|⨍|_\{\})) Mf(x) dx$ for all $⨍ ∈ L^\{1\}()$.},
author = {Kita, H.},
journal = {Studia Mathematica},
keywords = {Hardy-Littlewood maximal function; Orlicz space; inequality},
language = {eng},
number = {1},
pages = {1-10},
title = {On a converse inequality for maximal functions in Orlicz spaces},
url = {http://eudml.org/doc/216260},
volume = {118},
year = {1996},
}

TY - JOUR
AU - Kita, H.
TI - On a converse inequality for maximal functions in Orlicz spaces
JO - Studia Mathematica
PY - 1996
VL - 118
IS - 1
SP - 1
EP - 10
AB - Let $Φ(t) = ʃ_{0}^{t} a(s)ds$ and $Ψ(t) = ʃ_{0}^{t} b(s)ds$, where a(s) is a positive continuous function such that $ʃ_{1}^{∞} a(s)/s ds = ∞$ and b(s) is quasi-increasing and $lim_{s→∞}b(s) = ∞$. Then the following statements for the Hardy-Littlewood maximal function Mf(x) are equivalent: (j) there exist positive constants $c_1$ and $s_{0}$ such that $ʃ_{1}^{s} a(t)/t dt ≥ c_{1}b(c_{1}s)$ for all $s ≥ s_{0}$; (jj) there exist positive constants $c_2$ and $c_3$ such that $ʃ_{0}^{2π} Ψ((c_2)/(|⨍|_{}) |⨍(x)|) dx ≤ c_3 + c_{3} ʃ_{0}^{2π} Φ(1/(|⨍|_{})) Mf(x) dx$ for all $⨍ ∈ L^{1}()$.
LA - eng
KW - Hardy-Littlewood maximal function; Orlicz space; inequality
UR - http://eudml.org/doc/216260
ER -