Weighted inequalities for one-sided maximal functions in Orlicz spaces

Let ${\displaystyle M_g^{+}}$ be the maximal operator defined by ${\displaystyle M_g^{+}⨍\left(x\right)=\underset{h>0}{\text{sup}}\frac{{ʃ}_x^{x+h}|⨍|g}{{ʃ}_x^{x+h}g}}$, 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 ${\displaystyle {\Delta }_2}$. We characterize the pairs of positive functions (u,ω) such that the weak type inequality ${\displaystyle u\left(\{x\in ℝ\mid M_g^{+}⨍\left(x\right)>\lambda \}\right)\le \frac{C}{\Phi (\lambda )}{\int }_{ℝ}\Phi \left(|⨍|\right)\omega }$ holds for every ⨍ in the Orlicz space ${\displaystyle L_{\Phi }(\omega )}$. We also characterize the positive functions ω such that the integral inequality ${\displaystyle {\int }_{ℝ}\Phi \left(|M_g^{+}⨍|\right)\omega \le {\int }_{ℝ}\Phi \left(|⨍|\right)\omega }$ holds for every ${\displaystyle ⨍\in L_{\Phi }(\omega )}$. Our results include some already obtained for functions in ${\displaystyle L^p}$ and yield as consequences one-dimensional theorems due to Gallardo and Kerman-Torchinsky.