Weighted inequalities for one-sided maximal function in Orlicz classes

Abstract: Let $M_g^+$ be the one-sided maximal function and $\varphi$ be a nondecreasing finite function on $[0,\infty)$, $\varphi (0)=0$. Necessary and sufficient conditions are given on $\varphi$ and a weight function $w$ such that the integral inequalities of the form $$ \align \int_{-\infty}^{\infty}\varphi\big(M^+_gf(x)\big)w(x)g(x)dx& \leq c \int_{-\infty}^{\infty}\varphi(c|f(x)|)w(x)g(x)dx,
\int_{-\infty}^{\infty}\varphi\big(M^+_gf(x)w(x)\big)g(x)dx& \leq c \int_{-\infty}^{\infty}\varphi(c|f(x)|w(x))g(x)dx,
\intertext{and} \int_{-\infty}^{\infty}\varphi\Big(\frac{M^+_gf(x)}{w(x)}\Big)w(x)g(x)dx & \leq c \int_{-\infty}^{\infty}\varphi\Big(c\frac{|f(x)|}{w(x)}\Big)w(x)g(x)dx, \endalign $$ hold.

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