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In this paper, we will discuss the space of functions of weak bounded mean oscillation. In particular, we will show that this space is the dual space of the special atom space, whose dual space was already known to be the space of derivative of functions (in the sense of distribution) belonging to the Zygmund class of functions. We show, in particular, that this proves that the Hardy spaceH1{H}^{1}strictly contains the special atom space.

On the equivalence between weak BMO and the space of derivatives of the Zygmund class