Weyl sums in Fq[x] with digital restrictions

Let Fq be a finite field and consider the polynomial ring Fq [X]. Let Q ∈ Fq [X]. A function f : Fq [X] → G, where G is a group, is called strongly Q-additive, if f(AQ + B) = f(A) + f(B) holds for all polynomials A,B ∈ Fq [X] with degB < degQ. We estimate Weyl Sums in Fq [X] restricted by Q-additive functions. In particular, for a certain character E we study sums of the form X P E(h(P )), where h ∈ Fq((X))[Y ] is a polynomial with coefficients contained in the field of formal Laurent series over Fq and the range of P is restricted by conditions on fi(P ), where fi (1 ≤ i ≤ r) are Qi-additive functions. Adopting an idea of Gel fond such sums can be rewritten as sums of the form X degP<n E h(P ) + r X