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Interchange between weak Orlicz-Hardy spaces with concave functions through martingale transforms. (Interchange between weak Orlice-Hardy spaces with concave functions through martingale transforms.) (English) Zbl 1389.60059
Summary: In this paper, we consider the interchanging relation between two weak Orlicz-Hardy spaces associated with concave functions of martingales. By means of martingale transform, we prove that the elements in weak Orlicz-Hardy space \(w\mathcal H_{\Phi_1}\) are none other than the martingale transforms of those in \(w\mathcal H_{\Phi_2}\), where \(\Phi_1\) is a concave Young function, \(\Phi_2\) is a concave or a convex Young function and \(\Phi_1\preceq \Phi_2\) in some sense. It extends previous works in the literature from strong-type spaces to the setting of weak-type spaces, from norm inequalities to quasi-norm inequalities as well.
60G46 Martingales and classical analysis
60G42 Martingales with discrete parameter
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)