Stability of oscillatory integral asymptotics in two dimensions

The stability under phase perturbations of the decay rate of local scalar oscillatory integrals in two dimensions is analyzed. For a smooth phase S(x,y) and a smooth perturbation function f(x,y), the decay rate for phase S(x,y) + tf(x,y) is shown to be the same for all but finitely many t and given an explicit description. The decay rate of the generic S(x,y) + tf(x,y) is always at least as fast as that of S(x,y), and the "good" cases where it is the same as that of S(x,y) are explicitly described. Uniform stability of the decay rate is proven for S(x,y) + f(x,y) for small enough such good f(x,y), and the coefficient of the leading term of the asymptotics is shown to be Lipschitz of some order alpha, again for small enough good perturbations f(x,y).