These two families of functions can be characterized by their Walsh spectrum. A function $f$ from $\mathbb{F}_{p^{n}}$ to $\mathbb{F}_p$ is called an $s$-plateaued function if the Walsh transform $|\widehat{f}(\mu)|\in \{0,p^{\frac{n+s}{2}}\}$ for each $\mu \in\mathbb{F}_{p^n}$. A $0$-plateaued function $f$ is called as bent and its Walsh transform satisfies $|\widehat{f}(\mu)|= p^{\frac{n}{2}}$ for each $\mu \in\mathbb{F}_{p^n}$. Plateaued functions and bent functions play a significant role in cryptography, coding theory and sequences for communications [@Carlet1; @Carlet2; @CarM].

Boolean bent functions were introduced by Rothaus in [@rot1976]. These functions have optimal nonlinearity and can only exist when $n$ is even. In [@dillon1974], it is shown that the existence of Boolean bent functions is equivalent to the existence of a family of difference sets known as Hadamard difference sets. Boolean plateaued functions are introduced by Zheng and Zhang as a generalization of bent functions in [@ZZ]. Boolean plateaued functions have attracted the attention of researchers since these functions provide some suitable candidates that can be used in cryptosystems. A difference set characterization of these functions was recently provided by the second author. In [@olmez2], it is shown that the existence of Boolean plateaued functions is equivalent to the existence of partial geometric difference sets.

In arbitrary 