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A function $F:\mathbb{F}_{p^n}\rightarrow \mathbb{F}_{p^m},$ is a vectorial$s$-plateaued function if for each component function $F_{b}(\mu)=Tr_n(\alphaF(x)), b\in \mathbb{F}_{p^m}^*$ and $\mu \in \mathbb{F}_{p^n}$, the Walshtransform value $|\widehat{F_{b}}(\mu)|$ is either $0$ or $ p^{\frac{n+s}{2}}$.In this paper, we explore the relation between (vectorial) $s$-plateauedfunctions and partial geometric difference sets. Moreover, we establish thelink between three-valued cross-correlation of $p$-ary sequences and vectorial$s$-plateaued functions. Using this link, we provide a partition of$\mathbb{F}_{3^n}$ into partial geometric difference sets. Conversely, using apartition of $\mathbb{F}_{3^n}$ into partial geometric difference sets, weconstructed ternary plateaued functions $f:\mathbb{F}_{3^n}\rightarrow\mathbb{F}_3$. We also give a characterization of $p$-ary plateaued functionsin terms of special matrices which enables us to give the link between suchfunctions and second-order derivatives using a different approach.

Graphs of Vectorial Plateaued Functions as Difference Sets