Let \(\varphi :\ [0,1)\times [0,\infty )\rightarrow [0,\infty ]\) be a Musielak–Orlicz function and \(\gamma ,s\in (0,\infty )\) . In this article, the authors present some sufficient conditions to ensure that the dyadic maximal operator \(U_{\gamma ,s}\) is bounded on Musielak–Orlicz spaces \(L^{\varphi }[0,1).\) As applications, the authors establish the characterizations of Musielak–Orlicz Hardy spaces \(H_{\varphi }[0,1)\) and the boundedness of maximal Fejér operators from \(H_{\varphi }[0,1)\) to \(L^{\varphi }[0,1)\) . Furthermore, the almost everywhere convergence and the norm convergence of Fejér means of Walsh–Fourier series are also obtained. All these results include, as special cases, the essentially optimal conditions for variable exponent Lebesgue spaces, perturbed variable exponent Lebesgue spaces, and double-phase functionals with variable exponent Lebesgue spaces.
