Double Sums of Kloosterman Sums in Finite Fields

We bound double sums of Kloosterman sums over a finite field ${\mathbb F}_{q}$, with one or both parameters ranging over an affine space over its prime subfield ${\mathbb F}_p \subseteq {\mathbb F}_{q}$. These are finite fields analogues of a series of recent results by various authors in finite fields and residue rings. Our results are based on recent advances in additive combinatorics in arbitrary finite field.