Some Orlicz-norm inequalities for martingales

Martingale Orlicz‐Hardy spaces

“…As its applications we show the relation among five martingale Orlicz‐Hardy spaces and the duality, namely, the dual of martingale Orlicz‐Hardy spaces are generalized martingale Campanato spaces. The results on the relation among five martingale Orlicz‐Hardy spaces are generalizations of the results in 1, 5, 7. Further, we prove a John‐Nirenberg type inequality for martingales in generalized martingale Campanato spaces when the stochastic basis is regular.…”

“…To prove (2.7) we use (2.8). We omit its proof, since the method is the same as the proof of Theorem 3.5 in 5.…”

“…As an application of the atomic decomposition theorem, we prove the relation among five Hardy spaces, namely, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$P_{\Phi }=Q_{\Phi }\subset H_{\Phi }^s\subset H_{\Phi }^*\cap H_{\Phi }^S$\end{document} (Theorem 2.5). This relation is known by 5 when Φ is convex. If \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\lbrace \mathcal {F}_n\rbrace _{n\ge 0}$\end{document} is regular, then five Hardy spaces coincide, and consequently, the atomic decomposition theorem also holds for Hardy spaces \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H_{\Phi }^*$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H_{\Phi }^S$\end{document}.…”

Modular inequalities in martingale Orlicz–Karamata spaces

“…To prove , we use and . The method used below is the same as the proof of Theorem in . Assume that f is a martingale with .…”

“…To prove (4.8), we use (4.6) and (4.7). The method used below is the same as the proof of Theorem 3.5 in [28]. Assume that is a martingale with , [Φ( )] < ∞.…”