Martingale Transform And Hardy-lorentz Martingale Spaces

This paper mainly studies the analytical properties of Hardy-Lorentz martingale spaces.Firstly,the boundedness of martingale transform operator on Hardy-Lorentz martingale spaces and BMO spaces is proved.Secondly,by means of the technique of martingale transforms,it characterizes the relations between different predictable Hardy-Lorentz martingale spaces.At the end,according to the martingale transforms,the interchanging relations between Hardy-Lorentz martingale spaces and BMO spaces are investigated.The main content can be divided into by the following three parts.In the first part,it characterizes the boundedness of martingale transform operator.Taking advantage of the generalized Ho?lder's inequality in this part,the boundedness of martingale transform operator on Hardy-Lorentz martingale spaces and conjugate spaces is proved.At the second,making use of the technique of martingale transform,the interchanging relations between Hardy-Lorentz martingale spaces are characterized.The following conclusions are proved by the construction method and the K-method of interpolation spaces:let0<p₁<p₂<?,0<q<?,the martingale in Hardy-Lorentz spaceH_1p,_q can be represented as a martingale transform of some element in Hardy-Lorentz spaceH_p2,q,whereH_p-q?{?_p-q^s,?_p-q,Q_p-q}.Further research discusses the relationship betweenH_p1,q1andH_p2,q2for0<p₁<p₂<?and0<q₁<q₂<?,whereH_p-q?{?_p-q^s,?_p-q,Q_p-q}.As for the last part,making use of the technique of martingale transform,the characterizations of the interchanging relations between Hardy-Lorentz and BMO spaces are given.Let 0<p-q<?,the martingale in Hardy-Lorentz spaceH_p-q?{?_p-q^s,?_p-q,Q_p-q}can be represented as a martingale transform of some element in BMO?{BMO₂⁺,BMO₁,BMO₂}space.