Estimates for oscillatory integrals with phase having D type singularities

In this paper, we consider estimates for the two-dimensional oscillatory integrals. The phase function of the oscillatory integrals is the linear perturbation of a function having D type singularities. We consider estimates for the oscillatory integrals in terms of the Randol's type maximal functions. We obtain a sharp L^p_loc estimates for the Randol's maximal functions. Moreover, we investigate the sharp exponent p depending on whether, the phase function has linearly adapted coordinates system or not.