Sharp growth conditions for boundedness of maximal function in generalized Orlicz spaces

We study sharp growth conditions for the boundedness of the Hardy-Littlewood maximal function in the generalized Orlicz spaces. We assume that the generalized Orlicz function \(\phi(x,t)\) satisfies the standard continuity properties (A0), (A1) and (A2). We show that if the Hardy-Littlewood maximal function is bounded from the generalized Orlicz space to itself then \(\phi(x,t)/t^p\) is almost increasing for large \(t\) for some \(p>1\). Moreover we show that the Hardy-Littlewood maximal function is bounded from the generalized Orlicz space \(L^\phi(\mathbb{R}^n)\) to itself if and only if \(\phi\) is weakly equivalent to a generalized Orlicz function \(\psi\) satisfying (A0), (A1) and (A2) for which \(\psi(x,t)/t^p\) is almost increasing for all \(t>0\) and some \(p>1\). &nbsp