Martingale Hardy-Amalgam Spaces
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Date
2022-07
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Publisher
University of Ghana
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.
Description
PHd. In
Mathematics
Keywords
Spaces, Martingale Hardy-Amalgam