Martingale Hardy-Amalgam Spaces

Abstract

In this work, we introduce the new spaces, Hs p;q; HS p;q; H_ p;q; Qp;q; Pp;q; called the martingale Hardy-amalgam spaces. We study some of the properties of these newly introduced spaces; two de_nitions of atoms are given and hence two atomic decompositions are given, dualities of these spaces are characterized and the martingale inequalities and embeddings of these spaces are also discussed. It is proved that the dual of Hs p;q; (0 < p _ q _ 1); is a Campanato-type space and the dual of Hs p;q; (1 < p _ q < 1); is Hs p0;q0 where (p; p0); (q; q0) are conjugate pairs. The variation integrable space Gp;q is also introduced and it is established that the jump bounded space BDp;q is the dual of Gp;q: To be able to characterize this duality, a larger space, which we denote by K(Lp;q; `r); is introduced, such that Gp;q can be embedded into. The classical Doob's martingale inequality is also extended from the classical martingale Hardy spaces to the newly introduced martingale Hardy-amalgam spaces. The Burkholder-Davis-Gundy inequality is also extended from the classical martingale Hardy spaces to the martingale Hardy-amalgam spaces as well as the convexity inequality and the concavity inequalities involving measurable functions. The classical martingale Hardy space embeddings are also extended to the martingale Hardy-amalgam spaces. The Davis decompositions of martingales in the classical martingale Hardy spaces are also extended to the martingale Hardy-amalgam spaces. As an application of the Davis decomposition and the Garsia space, a duality theorem for H_ p;q (1 _ p; q _ 2) is provided. Finally, the boundedness of martingale transforms between the martingale Hardy-amalgam spaces are also discussed. No data was collected for this study as the methodology used is purely theoretical in nature.