On the Values of Kloosterman Sums

Given a prime $p$ and a positive integer $n$ , we show that the shifted Kloosterman sums $$sum _{x in BBF _{p^{n}}} psi (x + ax^{p^{n}-2}) = sum _{xin BBF _{p^{n}}^{ast }} psi(x + ax^{-1}) + 1, quad a inBBF _{p^{n}}^{ast }$$ where $psi$ is a nontrivial additive character of a finite field $BBF _{p^{n}}$ of $p^{n}$ elements, do not vanish if $a$ belongs to a small subfield $BBF_{p^{m}} subseteq BBF _{p^{n}}$. This complements recent results of P. Charpin and G. Gong which in turn were motivated by some applications to bent functions.