Estimates for oscillatory integrals with phase having $D$ type singularities

In this talk we consider estimates for the two-dimensional oscillatory integrals. The phase function of the oscillatory integral is a small perturbation of the function having $D$ type singularities. We consider estimates for the oscillatory integrals in terms of the generalized Randol’s type functions. We obtain the sharp estimate for the case when the phase has linearly adapted coordinates system. In the general case we get a bound which is close to the sharp bound.