Double sums of Kloosterman sums in finite fields

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We bound double sums of Kloosterman sums over a finite field Fq, with one or both parameters ranging over an affine space over its prime subfield Fp⊆Fq. 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.