Martingale Musielak–Orlicz–Lorentz Hardy Spaces with Applications to Dyadic Fourier Analysis

Abstract

Let $(\Omega ,{\mathcal {F}},{\mathbb {P}})$ be a probability space, $\varphi :\ \Omega \times [0,\infty )\rightarrow [0,\infty )$ a Musielak–Orlicz function, and $q\in (0,\infty ]$. In this article, the authors introduce five martingale Musielak–Orlicz–Lorentz Hardy spaces and prove that these new spaces have some important features such as atomic characterizations, the boundedness of $\sigma$-sublinear operators, and martingale inequalities. This new scale of martingale Hardy spaces requires the introduction of the Musielak–Orlicz–Lorentz space $L^{\varphi ,q}(\Omega )$. In particular, the authors show that this Lorentz type space has some fundamental properties including the completeness, the convergence, real interpolations, and the Fefferman–Stein vector-valued inequality for the Doob maximal operator. As applications, the authors prove that the maximal Fejér operator is bounded from the martingale Musielak–Orlicz–Lorentz Hardy space $H_{\varphi ,q}[0,1)$ to $L^{\varphi ,q}[0,1)$, which further implies some convergence results of the Fejér means. Moreover, all the above results are new even for Musielak–Orlicz functions with particular structure such as weight, weight Orlicz, and double-phase growth. The main approach used in this article can be viewed as a combination of the stopping time argument in probability theory and the real-variable technique of function spaces in harmonic analysis.