author provided a similar classification for weakly regular bent functions from $\mathbb{F}_3^n$ to $\mathbb{F}_3$ when $n$ is odd via partial geometric difference sets. Next we will also show that our vectorial functions have a similar classification in the case of $p=3$ and $d=\frac{3^{2k}+1}{2}$.

\[partitioned\]. Let $n \ge 3$ be an integer and $d=\frac{3^{2k}+1}{2}$ with $\gcd{(n,k)}=s$ and $n/s$ is odd. For $i=0,1,2$ the sets $D_i=\{x: F(x)=Tr_n(x^d)=i\}$ are $(3^n, 3^{n-1}, 3^{2n-3}-3^{n-2}, 3^{n-1}+3^{2n-3}-3^{n-2})$ partial geometric difference sets in the additive group of $\mathbb{F}_{3^{n}}$.

For each $a\in {{\mathbb F}}_{3^n}^*$, Suppose that $\chi_a(D_1)=x_a+y_a \zeta_3$ with $x_a,y_a\in \mathbb{R}$. Actually, when we are calculating $\chi_a(D_1)$, we are summing powers of $\zeta_3$ as many times as the number of elements in $D_1$ for which $Tr_{n}(ax)=0,1$ or $2$. In other words, if we set $$D_{1,i}=\{x\in D_1:Tr(x)=i\}, i=0,1,2$$ we have $$\chi_a(D_1)=|D_{1,0}|+|D_{1,1}|\zeta_3+|D_{1,2}|\zeta_3^2=(|D_{1,0}|-|D_{1,2}|)+(|D_{1,1}|-|D_{1,2}|)\zeta_3$$ and hence $$x_a=|D_{1,0}|-|D_{1,2}|,y_a=|D_{1,1}|-|D_{1,2}|$$ are both in $\mathbb{Z}$. Also note that for any $x \in \mathbb{F}_{3^n}$, $$F_1(-x)=Tr((-x)^d)=Tr(-x^d)=-F_1(x)$$ and that gives us $$x\in D_1 \Longleftrightarrow 2x \in D_2.$$ As a consequence, $\chi_a(D_2)=\overline{\chi_a(D_1)}=x_a+y_a \zeta_3^2$. Since $D_i$’s form a partition of the additive group of $\mathbb{F}_{3^{n}}$ $$\chi_a(D_0)+ \chi_a(D_1)+\chi_a(D_2)=0,$$ and we obtain $$\chi_a(D_0)=y_a-2x_a$$ Consider the Walsh transform values $\widehat{F_1}(a), \widehat{F_1}(-a)$ of $F_1(x)=Tr_n(x^d)$. $$\begin{aligned}
    \widehat{F_1}(-a)&=\sum_{x\in \mathbb{F}_{3^{n}}}\zeta_3^{Tr_{n}(ax+x^d)}\\
    &=\chi_a(D_0)+\zeta_3\chi_a(D_1)+\zeta_3^2\chi_a(D_2)\\
    &=y_a-2x_a+\zeta_3(x_a+y_a \zeta_3)+\zeta_3^2(x_a+y_a \zeta_3^2)\\
    &=y_a-2x_a+\zeta_3(x_a+y_a)+\zeta_3^2(x_a+y_a)=-3x_a\\
    \end{aligned}$$ and $$\begin{aligned}
    \widehat{F_1}(a)&=\sum_{x\in 