On the duality of some martingale spaces

Abstract

Fefferman has proved that the dual space of the martingale Hardy space H1 is the BMO1-space. Garsia went further and proved that the dual of Hp is the so-called martingale Kp-space, where p and q are two conjugate numbers and 1 ≤ p < 2.The martingale Hardy spaces HΦ with general Young function Φ, were investigated by Bassily and Mogyoródi. In this paper we show that the dual of the martingale Hardy space HΦ is the martingale Hardy space HΦ where (Φ, Ψ) is a pair of conjugate Young functions such that both Φ and Ψ have finite power. Moreover, two other remarkable dualities are presented.

Duality theorem for B‐valued martingale Orlicz–Hardy spaces associated with concave functions

“…All of them can also be found in the books of Long [11] and Weisz [18]. For Orlicz-Hardy spaces H s (2) (R), the dual space was obtained by Miyamoto, Nakai and Sadasue [14] when is a concave function and by Bassily and Abdel-Fattah [1] and Dam [2], [3] when is a convex function, respectively. As for the vector-valued case, introducing a new type of Garsia (B) for 0 < r ≤ 1 as a special case) still remains unresolved.…”

“…It is known that Orlicz spaces L and Orlicz-Hardy spaces H are very useful to know more precise properties of functions and martingales (see for example [1]- [3], [8], [14], [15], [21]- [23]). In martingale theory, martingale Orlicz-Hardy spaces H associated with concave functions , which are the generalizations of Hardy spaces H p for 0 < p ≤ 1, were introduced recently by Miyamoto, Nakai and Sadasue [14] and also were investigated by Jiao and Yu [8].…”