It is well-known that the Hardy spaces H^{p}(R^{n}) are good substitutes for Lebesgue spaces L^{p}(R^{n}) with 0<p≤1. Moreover, when it comes to studying the boundedness of operators in the critical case, weak Hardy spaces WH^{p}(R^{n}) naturally appear, and it is proven to be a good substitute of Hardy spaces H^{p}(R^{n}) with 0<p≤1.

For example, if 0<δ≤1, δ-Calderón-Zygmund operator T is bounded on H^{p}(R^{n}) for any n/(n+δ)<p≤1, but T may be not bounded on H^{n/(n+}^{δ}^{)}(R^{n}); however, T is bounded by Hardy space H^{n/(n+}^{δ}^{)}(R^{n}) to weak Hardy space WH^{n/(n+}^{δ}^{)}(R^{n}).

On the other hand, as a natural generalization of L^{p}(R^{n}), the Orlicz space was introduced by Birnbaum and Orlicz. Orlicz space can be further generalized to Musielak-Orlicz space which may also vary in the spatial variables. Musielak-Orlicz spaces include many function spaces far beyond L^{p}(R^{n}), and its motivation comes from various applications to mathematics and physics.

Anisotropic phenomena appear in many aspects of mathematical analysis and its applications. Anisotropic function spaces on R^{n} have been extensively studied, beginning with the Russian school in the 1960s, and then M. Bownik, K.P-Ho, D. Yang, and J. Liu, etc.

This article introduces the anisotropic weak Hardy spaces of the Musielak-Orlicz type. These kinds of spaces are appropriate general spaces which include weak Hardy spaces of R. Fefferman and F. Soria, weighted weak Hardy spaces of T. Quek and D. Yang, and the anisotropic weak Hardy space of Y. Ding and S. Lan.

Next, the atomic characterization of this space is also obtained. That is, every element in this space can be represented as a sum of countable, infinite functions which has better properties in the sense of distribution. Here, we point out these infinite functions are called atoms. These atoms have compact supports, size conditions, and vanishing conditions. Therefore, when we tried to obtain some boundedness of linear operators on the elements of the anisotropic weak Hardy spaces of Musielak-Orlicz type, through some additional proper conditions, it can be transferred to the linear operators act on these atoms, and with those better properties on atoms, this will make the estimate of T on the space more convenient.

To be precise, this article gives two examples. In the first example, an interpolation theorem adapted to this space was obtained. This result extends the corresponding conclusion of Y. Ding and S. Lan in 2008, who obtained the interpolation on weak anisotropic Hardy spaces. Here, we point out that, since the general weak Hardy spaces don’t have their corresponding dense subspaces, we have to use a new superposition principle adapted to the weak anisotropic Musielak-Orlicz function spaces. This is an extension of the corresponding conclusion from E. M. Stein, M. Taibleson, G. Weiss in 1981.

Moreover, as another application of the atomic decomposition of this space, the boundedness of anisotropic Calderón-Zygmund operators from anisotropic weak Hardy space of Musielak-Orlicz type to anisotropic weak Lebesgue space of Musielak-Orlicz type is also established. This result also extends the corresponding result of Y. Ding and S. Lan, who obtained the anisotropic Calderón-Zygmund operators from anisotropic weak Hardy space H^{1} to anisotropic weak Lebesgue space L^{1} in 2008. The result is new even when it is reduced to weighted weak Orlicz Hardy spaces.

These findings are described in the article entitled Anisotropic weak Hardy spaces of Musielak-Orlicz type and their applications, recently published in the journal *Frontiers of Mathematics in China*. This work was conducted by Hui Zhang, Chunyan Qi, and Baode Li from Xinjiang University.