  For a small disk D centered at the origin in R^2, a smooth real-valued
function S(x,y) on D, and a positive epsilon, we consider the measure of the
points in D where |S(x,y)| < epsilon, as well as oscillatory integral
analogues. Specifically, we consider the effect of perturbing S(x,y) on these
quantities. Besides being of intrinsic interest, these questions are important
in the analysis of Fourier transforms of surface-supported measures. Complex
and higher-dimensional analogues of these questions are also connected to
various issues in algebraic and complex geometry. For real-analytic S(x,y),
this question has been investigated for example by Karpushkin, using versal
deformation theory, and by Phong-Stein-Sturm, who developed a method often
referred to as the method of algebraic estimates.
  In this paper, we show how the use of resolution of singularities algorithms
in two dimensions, along with some one-dimensional Van der Corput-type lemmas,
provides another method for dealing with such questions. As a result, we prove
new estimates and theorems for these and related quantities. Furthermore, since
these algorithms apply to all smooth functions, the theorems will hold for all
smooth functions as opposed to the earlier real-analytic results.
