Lépingle inequality and martingale Hardy spaces

Abstract

We prove that if $(\Omega ,\mathcal{F},\mathbb{P})$ is a probability space equipped with a filtration $({\mathcal{F}}_n{)}_{n\ge 1}$ and $E(\Omega )$ is a quasi-Banach Köthe function space with the property that the Lépingle inequality is satisfied for adapted sequences in $E(\Omega )$, then the couple of martingale Hardy spaces $(H_E^S(\Omega ),H_{\infty }^S(\Omega ))$ is $K$-closed in the couple of Köthe–Bochner spaces $(E(\Omega ;{\ell }_2),L_{\infty }(\Omega ;{\ell }_2))$. This extends the commutative form of a recent result of Moyart from $L_1$ 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 $(E(\Omega ),L_{\infty }(\Omega ))$ to the couple of corresponding conditional martingale Hardy spaces $(H_E^s(\Omega ),H_{\infty }^s(\Omega ))$.