Boundedness of Dyadic Maximal Operators on Musielak-Orlicz Type Spaces and Its Applications

Let phi:[0,1)x[0,infinity)->[0,infinity]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi :\ [0,1)\times [0,\infty )\rightarrow [0,\infty ]$$\end{document} be a Musielak-Orlicz function and gamma,s is an element of(0,infinity)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma ,s\in (0,\infty )$$\end{document}. In this article, the authors present some sufficient conditions to ensure that the dyadic maximal operator U gamma,s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U_{\gamma ,s}$$\end{document} is bounded on Musielak-Orlicz spaces L phi[0,1).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>{\varphi }[0,1).$$\end{document} As applications, the authors establish the characterizations of Musielak-Orlicz Hardy spaces H phi[0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\varphi }[0,1)$$\end{document} and the boundedness of maximal Fej & eacute;r operators from H phi[0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_{\varphi }[0,1)$$\end{document} to L phi[0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>{\varphi }[0,1)$$\end{document}. Furthermore, the almost everywhere convergence and the norm convergence of Fej & eacute;r 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.