Weighted inequalities for one-sided maximal functions in Orlicz spaces
Volume 131 / 1998
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.
