On maximal functions in Orlicz spaces

Abstract:

Let $\Phi (t)$ and $\Psi (t)$ be the functions having the representations $\Phi (t)=\int _{0}^{t} a(s)ds$ and $\Psi (t)=\int _{0}^{t} b(s)ds$, where $a(s)$ is a positive continuous function such that $\int _{1}^{\infty }\frac {a(s)}{s}ds=+\infty$ and $b(s)$ is quasi-increasing. Then the maximal function $Mf$ is a function in Orlicz space $L^{\Phi }$ for all $f\in L^{\Psi }$ if and only if there exists a positive constant $c_{1}$ such that $\int _{1}^{s} \frac {a(t)}{t}dt\leq c_{1}b(c_{1}s)$ for all $s\geq 1$.