Weak atomic decompositions of B-valued martingales with two-parameters in weak Hardy spaces wpΣα and wpHα are established and the boundedness of sublinear operators on these spaces are proved. By using them, some characterizations
of the smoothness of Banach spaces are obtained.
The phenomenon of all-elements-sleeping-beauties in science is revealed by four special cases. The ‘sleeping beauties’ prick their fingers on the ‘spindles’ so that they fall into sleep then are awakened by their ‘princes’. The authors speculate that the phenomenon could happen in scientific literatures with high quality.