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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… Click to show full abstract

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 } .

Keywords: predictable martingales; lorentz spaces; interchange hardy; spaces predictable; hardy lorentz; space

Journal Title: Journal of Mathematical Analysis and Applications
Year Published: 2019

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