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Martingale Orlicz-Hardy spaces. (English) Zbl 1260.60082

Suppose \(\left( \Omega, \mathcal{F}, P \right)\) is a probability space. Let \(\mathcal{M}\) be the set of all martingales \(f=\{f_n\}_{n\geqslant 0}\) relative to a non-decreasing sequence of \(\sigma\)-subalgebras \(\{\mathcal{F}_n\}_{n\geqslant 0}\) generating the \(\sigma\)-algebra \(\mathcal{F}\). For a martingale \(f \in \mathcal{M}\), the authors define the maximal function \(f^*\), the quadratic variation \(S(f)\), and the conditional quadratic variation \(s(f)\). Let \(L_\Phi\) be the Orlicz space associated with a non-convex Orlicz function \(\Phi\) satisfying certain additional conditions. Then, the authors use \(f^*\), \(S(f)\) and \(s(f)\) to define five martingale Hardy-Orlicz subspaces of \(\mathcal{M}\). Their results include an atomic decomposition for some of these Hardy-Orlicz spaces. In particular, if the sequence of \(\sigma\)-algebras \(\{\mathcal{F}_n\}_{n\geqslant 0}\) is a regular stochastic basis, then all these five spaces coincide, and so one gets an atomic decomposition for all five spaces. As another application of the atomic decomposition, the authors show that the dual of one of these spaces is a generalized martingale Campanato space \(\mathcal{L}_{2,\phi}\) with a certain relation between functions \(\Phi\) and \(\phi\). Finally, a John-Nirenberg-type inequality for \(\mathcal{L}_{2,\phi}\) is proved under the assumption that \(\{\mathcal{F}_n\}_{n\geq 0}\) is regular.

MSC:

60G46 Martingales and classical analysis
42B30 \(H^p\)-spaces
42B35 Function spaces arising in harmonic analysis
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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References:

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