  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).
