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Let $\mathscr{F}$ be a formation and $G$ a finite group. The weak norm of asubgroup $H$ in $G$ with respect to $\mathscr{F}$ is defined by$N_{\mathscr{F}}(G,H)=\underset{T\leq H}{\bigcap}N_G(T^{\mathscr{F}})$. Inparticular, $N_{\mathscr{F}}(G)=N_{\mathscr{F}}(G,G)$. Let$N^i_{\mathscr{F}}(G)$,$i\geq 1$, be a upper series of $G$ by setting$N^0_{\mathscr{F}}(G)=1$,$N^{i+1}_{\mathscr{F}}(G)/N^i_{\mathscr{F}}(G)=N_{\mathscr{F}}(G/N^i_{\mathscr{F}}(G))$and denoted by $N^{\infty}_{\mathscr{F}}(G)$ the terminal term of the series.In this paper, for the case $\mathscr{F}\in\{\mathscr{U}_p,\mathscr{U}\}$,where $\mathscr{U}_p$($\mathscr{U}$,respectively) is the class of all finite$p$-supersolvable groups(supersolvable groups,respectively), we characterizethe structure of some given finite groups by the properties of weak norm ofsome subgroups in $G$ with respect to $\mathscr{F}$. Some of our main resultsmay regard as a continuation of many nice previous work.

On the weak norm of $\mathscr{U}_p$-residuals of all subgroups of a finite group