Martingale transforms between Hardy–Lorentz spaces

Abstract

Using the technique of Burkholder’s martingale transforms, the interchanging relations between two Hardy–Lorentz spaces of martingales are characterized. More precisely, it is proved that the elements in Hp1,qs (resp. Hp1,q1s) are none other than the martingale transforms of those in Hp2,qs (resp. Hp2,q2s), when 0<p1<p2<∞, 0<q<∞ (resp. 0<p1<p2<∞, 0<q1<q2<∞ and p1p2=q1q2), and a martingale is in Hp,qs for 1<p,q<∞ if and only if it is a martingale transform of some martingale from BMO2.