is no solution to $u\pi(y)=a^{2^i}$. Hence $W_{f}(a,b)=0$.

[**B. More on $(n,n)$-functions**]{}

In [@Pott], Pott et al. presented in univariate form an infinite class of $(2k,2k)$-functions $x^{2^{i}}(x+x^{2^{k}})$ with the maximal number of bent components. They noticed that these functions are of the form $F(x)=x^{2^{i}}\mathcal{L}(x)$ for some linear mapping $\mathcal{L}(x)=x+x^{2^{k}}$. Then they raised an open question (Question 2 in [@Pott]) of determining all those linear mappings $\mathcal{L}$ over $\mathbb{F}_{2^{2k}}$ such that the function $x\mathcal{L}(x)$ has the maximal number of bent components. In the following, we completely determine under which conditions a large subclass of such $(n,n)$-functions will have the maximum number of bent components, and thus partially answer the open question. [ ]{}

It should be also noted that the $(n,n)$-functions constructed by [@MZTZ; @Pott] are all quadratic. The proofs for that those functions have the maximum number of bent components are long, and they rely on the theory of quadratic forms. It seems that to find such functions, perhaps one has to examine the quadratic functions first. However, in this paper, we break this impression and provide a direct and much simpler proof for the maximality of the number of bent components of the functions considered in the following theorem, which 