On subspaces of Kloosterman zeros and permutations of the form $L_1(x^{-1})+L_2(x)$

Permutations of the form $F=L_1(x^{-1})+L_2(x)$ with linear functions $L_1,L_2$ are closely related to several interesting questions regarding CCZ-equivalence and EA-equivalence of the inverse function. In this paper, we show that $F$ cannot be a permutation if the kernel of $L_1$ or $L_2$ is too large. A key step of the proof is a new result on the maximal size of a subspace of $\mathbb{F}_{2^n}$ that contains only Kloosterman zeros, i.e. a subspace $V$ such that $K_n(v)=0$ for all $v \in V$ where $K_n(v)$ denotes the Kloosterman sum of $v$.}