Weights of twisted exponential sums

Let k be a finite field of characteristic p, l a prime number different from p, psi : k -> (Q) over bar (l)* a nontrivial additive character, and chi : k*(n) -> (Q) over bar (l)* a character on k*(n). Then psi defines an Artin-Schreier sheaf L(psi) on the affine line A(k)(1), and chi defines a Kummer sheaf K(chi) on the n-dimensional torus T(k)(n). Let f is an element of k[X(1), X(1)(-1), ... , X(n) , X(n)(-1)] be a Laurent polynomial. It defines a k-morphism f : T(k)(n) -> A(k)(1). In this paper, we calculate the weights of H(c)(i) (T((k) over bar)(n), K(chi) circle times f* L(psi)) under some non-degeneracy conditions on f. Our results can be used to estimate sums of the form Sigma(x1, ... , xn is an element of k*) chi 1(f(1)(x(1), ... , x(n))) ... chi(m) (f(m) (x(1), ..., x(n)))psi(f(x(1), ... , x(n))), where chi(1), ... , chi(m) : k* -> C* are multiplicative characters, psi : k -> C* is a nontrivial additive character, and f(1), ... , f(m), f are Laurent polynomials.