Interchange between Hardy–Lorentz spaces of predictable martingales

Abstract

By means of the technique of Burkholder's martingale transforms, the interchanging relations between “predictable” martingale Hardy–Lorentz spaces are characterized. More precisely, let 0 < p 1 < p 2 < ∞ and 0 < q 1 ≤ q 2 < ∞ , it is shown that the elements in Hardy–Lorentz space H p 1 , q 1 are none other than the martingale transforms of those in Hardy–Lorentz space H p 2 , q 2 , where H p i , q i ∈ { P p i , q i , Q p i , q i } for i = 1 , 2 . At the endpoint case, the space H ∞ must be replaced by a B M O space, it is also proved that a martingale is in H p , q ∈ { P p , q , Q p , q } for 0 < p , q < ∞ , if and only if it is the transform of a martingale from B M O ∈ { B M O 1 , B M O 2 } .