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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.
##### MSC:
 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.)