New Martingale Inequalities and Applications to Fourier Analysis

Abstract

Let $(\Omega ,\mathcal{F},\mathbb{P})$ be a probability space and $\phi :\Omega \times [0,\infty )\to [0,\infty )$ be a Musielak-Orlicz function. In this article, the authors prove that the Doob maximal operator is bounded on the Musielak-Orlicz space $L^{\phi }(\Omega )$. Using this and extrapolation method, the authors then establish a Fefferman-Stein vector-valued Doob maximal inequality on $L^{\phi }(\Omega )$. As applications, the authors obtain the dual version of the Doob maximal inequality and the Stein inequality for $L^{\phi }(\Omega )$, which are new even in weighted Orlicz spaces. The authors then establish the atomic characterizations of martingale Musielak-Orlicz Hardy spaces $H_{\phi }^s(\Omega )$, $P_{\phi }(\Omega )$, $Q_{\phi }(\Omega )$, $H_{\phi }^S(\Omega )$ and $H_{\phi }^M(\Omega )$. From these atomic characterizations, the authors further deduce some martingale inequalities between different martingale Musielak-Orlicz Hardy spaces, which essentially improve the corresponding results in Orlicz space case and are also new even in weighted Orlicz spaces. By establishing the Davis decomposition on $H_{\phi }^S(\Omega )$ and $H_{\phi }^M(\Omega )$, the authors obtain the Burkholder-Davis-Gundy inequality associated with Musielak--Orlicz functions. Finally, using the previous martingale inequalities, the authors prove that the maximal Fejér operator is bounded from $H_{\phi }[0,1)$ to $L^{\phi }[0,1)$, which further implies some convergence results of the Fejér means; these results are new even for the weighted Hardy spaces.

58 pages; Submitted