Decay estimates for oscillatory integrals with polynomial phase. (Q1873674)

The paper deals with oscillatory integrals \(\int^b_a e^{i\lambda \varphi(t)}dt\) with phase \(p(t)\) being a real monic polynomial of degree \(n\) and big positive parameter \(\lambda\). It is supposed that for some \(j\), \(1\leq j\leq n-1\), the \(j\)-th derivative \(p^{(j)}(t)\) of \(p(t)\) has real and simple roots only. There are obtained upper estimates of the oscillatory integral by the value \(C_n\lambda^{-1/(j+1)}\), where \(C_n\) depends on the difference between the roots of \(p^{(j)}\) and \(p^{(j+1)}\). Similar estimates are also obtained for the case of multiple real roots of \(p^{(j)}\) when the \(j\)-th derivative has all roots real and some multiple. The sharpness of the results is discussed. Cf. also the two articles by \textit{D. Oberlin} [Proc. Am. Math. Soc. 125, No. 5, 1355--1361 (1997; Zbl 0972.42006) and Math. Res. Lett. 5, No. 5, 773--776 (1997; Zbl 0922.42010)].