Weak martingale Hardy spaces and weak atomic decompositions

Abstract

In this paper we define some weak martingale Hardy spaces and three kinds of weak atoms. They are the counterparts of martingale Hardy spaces and atoms in the classical martingale Hp-theory. And then three atomic decomposition theorems for martingales in weak martingale Hardy spaces are proved. With the help of the weak atomic decompositions of martingale, a sufficient condition for a sublinear operator defined on the weak martingale Hardy spaces to be bounded is given. Using the sufficient condition, we obtain a series of martingale inequalities with respect to the weak Lp-norm, the inequalities of weak (p , p)-type and some continuous imbedding relationships between various weak martingale Hardy spaces. These inequalities are the weak versions of the basic inequalities in the classical martingale Hp-theory.

New weak martingale Hardy spaces of Musielak–Orlicz type

“…The weak Hardy space was originally introduced by Fefferman and Soria [4], and then undergone a vast research, see, for example, [3,7,18]. Meanwhile, as a counterpart to the Hardy spaces of functions, the martingale Hardy spaces and weak martingale Hardy spaces were also studied by many authors, see, for example, [6,9,16,22,23]. We know that the atomic decompositions of the Hardy spaces H p (R n ), which were obtained by Coifman [1] when n = 1 and Latter [14] when n > 1, are very important in the real-variable theory.…”

“…Since then, atomic decompositions were established for many other martingale spaces, see [9] for weak martingale Hardy spaces, [10,12] for martingale Hardy-Lorentz spaces, [17] for Orlicz-Hardy martingale spaces, [11] for weak Orlicz-Hardy martingale spaces, [24] for weak Orlicz-Lorentz martingale spaces, [8] for Lorentz-Karamata martingale spaces, and so on.…”

Interpolation for weak Orlicz spaces with M Δ condition

“…In this paper we discuss operator interpolation for one class of weak Orlicz spaces generalized by N-functions satisfying M Δ condition, which was firstly introduced by CaJIeXoB in 1968. Then we show that it is also true in martingale setting, and the new technique of weak atomic decomposition (see [8]) plays an important role in our proof (we think that applying the new technique to interpolation theory is the main contribution of the paper). By the interpolation theorem it is easier to investigate the embedding relationships among weak Orlicz martingale spaces involving two N-functions.…”

Weak atomic decomposition of B-valued martingales with two-parameters

“…Weisz [4], [12] also made further studies of the atomic decompositions for the weak Hardy spaces consisting of the Vilenkin martingales and proved a weak version of the Hardy-Littlewood inequality. Recently, Liu and Hou [8] introduced the p-atom for B-valued martingales and studied the atomic decompositions in B-valued martingale spaces; Hou and Ren [9] studied the weak atomic decompositions for weak martingales in one-parameter case; Cheng and Gan [10] obtained atomic decompositions for B-valued martingales in two-parameter case. It is well-known that there are close interrelations between the geometrical properties of Banach spaces and the theory of B-valued martingales [5,6].…”