  A function $F:\mathbb{F}_2^n\rightarrow \mathbb{F}_2^n$, $n=2m$, can have at
most $2^n-2^m$ bent component functions. Trivial examples are obtained as $F(x)
= (f_1(x),\ldots,f_m(x),a_1(x),\ldots, a_m(x))$, where
$\tilde{F}(x)=(f_1(x),\ldots,f_m(x))$ is a vectorial bent function from
$\mathbb{F}_2^n$ to $\mathbb{F}_2^m$, and $a_i$, $1\le i\le m$, are affine
Boolean functions. A class of nontrivial examples is given in univariate form
with the functions $F(x) = x^{2^r}{\rm Tr^n_m}(\Lambda(x))$, where $\Lambda$ is
a linearized permutation of $\mathbb{F}_{2^m}$. In the first part of this
article it is shown that plateaued functions with $2^n-2^m$ bent components can
have nonlinearity at most $2^{n-1}-2^{\lfloor\frac{n+m}{2}\rfloor}$, a bound
which is attained by the example $x^{2^r}{\rm Tr^n_m}(x)$, $1\le r<m$ (Pott et
al. 2018). This partially solves Question 5 in Pott et al. 2018. We then
analyse the functions of the form $x^{2^r}{\rm Tr^n_m}(\Lambda(x))$. We show
that for odd $m$, only $x^{2^r}{\rm Tr^n_m}(x)$, $1\le r<m$, has maximal
nonlinearity, whereas there are more of them for even $m$, of which we present
one more infinite class explicitly. In detail, we investigate Walsh spectrum,
differential spectrum and their relations for the functions $x^{2^r}{\rm
Tr^n_m}(\Lambda(x))$. Our results indicate that this class contains many
nontrivial EA-equivalence classes of functions with the maximal number of bent
components, if $m$ is even, several with maximal possible nonlinearity.
