Remark on the formula for $L^p$ norm characterized by weak $L^p$ norm

In \cite{s=0}, the new formula for $L^p({\mathbb{R}}^N)$ norms was given by Gu and Yung which is characterized by the standard weak-$L^p({\mathbb{R}}^{2N})$ norms. At first glance, it seems to be able to measure the degree of fractional smoothness of functions. In this paper, our proof goes in parallel as in \cite{s=0}, but gives another different representation which is obviously independent of the smoothness of function.