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September 10, 2025Open Access

Lépingle inequality and martingale Hardy spaces

NRNarcisse RandrianantoaninaMiami University

Key Points

The proof shows that martingale Hardy spaces are K-closed in Köthe–Bochner spaces under the Lépingle inequality framework.
Key results extend the commutative form of the earlier findings from L1 to general Köthe function spaces.
Applications include new interpolation results for Musielak–Orlicz and variable martingale Hardy spaces.
An analogous result on automatic transfer of real interpolation from quasi-Banach Köthe function spaces to conditional martingale Hardy spaces is established.

Abstract

We prove that if (, F, P) is a probability space equipped with a filtration (F₍) ₍ ₁ and E () is a quasi-Banach Köthe function space with the property that the Lépingle inequality is satisfied for adapted sequences in E (), then the couple of martingale Hardy spaces (H₄^S (), H_^S () ) is K -closed in the couple of Köthe–Bochner spaces (E (;₂), L_ (;₂) ). This extends the commutative form of a recent result of Moyart from L₁ to general Köthe function spaces and provides a lifting of real interpolation of function spaces to corresponding martingale Hardy spaces. As applications, we obtain new type of interpolation results for Musielak–Orlicz martingale Hardy spaces and variable martingale Hardy spaces. We also prove an analogous result on automatic transfer (without any assumption) of real interpolation of couple of quasi-Banach Köthe function spaces (E (), L_ () ) to the couple of corresponding conditional martingale Hardy spaces (H₄^s (), H_^s () ).

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