  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 $v$.}
