Weighted inequalities for one-sided maximal functions in Orlicz spaces

Ortega Salvador, Pedro. "Weighted inequalities for one-sided maximal functions in Orlicz spaces." Studia Mathematica 131.2 (1998): 101-114. <http://eudml.org/doc/216567>.

@article{OrtegaSalvador1998,
abstract = {Let $M_\{g\}^\{+\}$ be the maximal operator defined by $M_\{g\}^\{+\}⨍(x) = sup_\{h>0\} (ʃ_\{x\}^\{x+h\} |⨍|g)/(ʃ_\{x\}^\{x+h\} g)$, where g is a positive locally integrable function on ℝ. Let Φ be an N-function such that both Φ and its complementary N-function satisfy $Δ_2$. We characterize the pairs of positive functions (u,ω) such that the weak type inequality $u(\{x ∈ ℝ | M_\{g\}^\{+\}⨍(x) > λ\}) ≤ C/(Φ(λ)) ʃ_ℝ Φ(|⨍|)ω$ holds for every ⨍ in the Orlicz space $L_Φ(ω)$. We also characterize the positive functions ω such that the integral inequality $ʃ_ℝ Φ(|M_\{g\}^\{+\}⨍|)ω ≤ ʃ_ℝ Φ(|⨍|)ω$ holds for every $⨍ ∈ L_Φ(ω)$. Our results include some already obtained for functions in $L^p$ and yield as consequences one-dimensional theorems due to Gallardo and Kerman-Torchinsky.},
author = {Ortega Salvador, Pedro},
journal = {Studia Mathematica},
keywords = {one-sided maximal functions; weighted inequalities; weights; Orlicz spaces; Hardy-Littlewood maximal function; weak-type estimates},
language = {eng},
number = {2},
pages = {101-114},
title = {Weighted inequalities for one-sided maximal functions in Orlicz spaces},
url = {http://eudml.org/doc/216567},
volume = {131},
year = {1998},
}

TY - JOUR
AU - Ortega Salvador, Pedro
TI - Weighted inequalities for one-sided maximal functions in Orlicz spaces
JO - Studia Mathematica
PY - 1998
VL - 131
IS - 2
SP - 101
EP - 114
AB - Let $M_{g}^{+}$ be the maximal operator defined by $M_{g}^{+}⨍(x) = sup_{h>0} (ʃ_{x}^{x+h} |⨍|g)/(ʃ_{x}^{x+h} g)$, where g is a positive locally integrable function on ℝ. Let Φ be an N-function such that both Φ and its complementary N-function satisfy $Δ_2$. We characterize the pairs of positive functions (u,ω) such that the weak type inequality $u({x ∈ ℝ | M_{g}^{+}⨍(x) > λ}) ≤ C/(Φ(λ)) ʃ_ℝ Φ(|⨍|)ω$ holds for every ⨍ in the Orlicz space $L_Φ(ω)$. We also characterize the positive functions ω such that the integral inequality $ʃ_ℝ Φ(|M_{g}^{+}⨍|)ω ≤ ʃ_ℝ Φ(|⨍|)ω$ holds for every $⨍ ∈ L_Φ(ω)$. Our results include some already obtained for functions in $L^p$ and yield as consequences one-dimensional theorems due to Gallardo and Kerman-Torchinsky.
LA - eng
KW - one-sided maximal functions; weighted inequalities; weights; Orlicz spaces; Hardy-Littlewood maximal function; weak-type estimates
UR - http://eudml.org/doc/216567
ER -