A general bound for oscillatory integrals with a polynomial phase of degree k

Let f is an element of R[X-1,...,X-n] be a polynomial of degree k >= 2. We consider the oscillatory integral I(lambda) = integral phi(x)e(i lambda f(x))dx, where phi is a C-1 function with compact support. A classical result due to E.M. Stein implies that I(lambda) = O(lambda(-1/k)), as lambda --> +infinity. The exponent 1/k is best possible, as shown by the example f(x) = f (x(0))+/-L(x - X-0)(k), where x(0) is any point in R-n and L is any nonzero linear form on R-n. In this paper, we show that, if f is precisely not of the above form, then the stronger bound I(lambda) O(lambda(-1/(k-1))) holds, and the exponent -1/(k - 1) is best possible.