Decay estimates for oscillatory integrals with polynomial phase in terms of p(n−3) and p(n−2) or in terms of p(n−2) and p(n−1)

We obtain estimates for certain oscillatory integrals with polynomial (degree n) phase, p(t). These estimates are stated in terms of differences between the roots, real or complex, of p(n−3)(t ) and p(n−2)(t ) or between p(n−2)(t ) and p(n−1)(t ). The sharpness of these results is also explored. This result is a partial generalization of the results found in [J. Math. Anal. Appl. 280 (2003) 424].  2004 Elsevier Inc. All rights reserved.