It is known that the dual of a weakly regular bent function is again weakly regular. On the other hand, the dual of a non-weakly regular bent function may not even be a bent function. In 2013, Çesmelioğlu, Meidl and Pott pointed out that the existence of a non-weakly regular bent function having weakly regular bent dual is an open problem. In this paper, we prove that for an odd prime p and $n\in {\mathbb{Z}}^{+}$, if $f:{\mathbb{F}}_p^n\to {\mathbb{F}}_p$ is a non-weakly regular bent function such that its dual $f^{⁎}$ is bent, then $f^{⁎⁎}(-x)=f(x)$, and $f^{⁎}$ is non-weakly regular, which solves the open problem. We also generalize our results to plateaued functions.

中文翻译：

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非弱正则弯曲函数的对偶不是弱正则函数并且推广到平稳函数

众所周知，弱规则弯曲函数的对偶也是弱规则的。另一方面，非弱规则弯曲函数的对偶甚至可能不是弯曲函数。Çesmelioğlu，Meidl和Pott在2013年指出，存在具有弱规则弯曲对偶的非弱规则弯曲函数是一个未解决的问题。在本文中，我们证明对于奇数素数p和$ñ\in {\mathbb{ž}}^{+}$如果 $F：{\mathbb{F}}_p^{ñ}\to {\mathbb{F}}_p$ 是非弱正则弯曲函数，因此其双重 $F^{⁎}$ 弯曲，然后 $F^{⁎⁎}（-X）=F（X）$和 $F^{⁎}$是非弱规则的，这解决了开放问题。我们还将结果推广到稳定的功能。