Connections between quaternary and component Boolean bent functions

This paper is about quaternary bent functions. Function $g:\mathbb{Z}_4^n\rightarrow\mathbb{Z}_4$ is called quaternary in $n$ variables. It was proven that bentness of a quaternary function $g(x+2y)=a(x,y)+2b(x,y)$ doesn't directly depend on the bentness of Boolean functions $b$ and $a\oplus b$. The number of quaternary bent functions in one and two variables is obtained with a description of properties of Boolean functions $b$ and $a\oplus b$. Two simple constructions of quaternary bent functions in any number of variables are presented. The first one is given by the formula $g(x_1+2x_{n+1},\ldots,x_n+2x_{2n})=\sum\limits_{i=1}^n2x_ix_{i+n} + cx_j$, $c\in\mathbb{Z}_2$ and $j\in\{1,\ldots,n\}$. The second construction allows one to get a bent function $g'(x+2y)=3a(x,y) + 2b(x,y)$, where $g(x+2y)=a(x,y) + 2b(x,y)$ is bent.