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In \cite{s=0}, the new formula for $L^p(\mathbb{R}^N)$ norms was given by Guand 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 fractionalsmoothness of functions. In this paper, our proof goes in parallel as in\cite{s=0}, but gives another different representation which is obviouslyindependent of the smoothness of function.

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