Uniform estimates of oscillatory integrals with phase from the series \(\widetilde R_m\) (Q1290823)

The author proves that \(C\tau^{-1}(\varphi)_2\) is a sharp uniform upper estimate of the absolute value of an oscillatory integral with a large parameter \(\tau\) and a phase from the series \(\widetilde R_m\), i.e., \(\widetilde T_{3,m,m}\); it has been known earlier that such an integral has a uniform upper estimate of \(C\tau^{-1}\ln\tau(\varphi)_2\). Here \((\varphi)_k\) is the norm of the amplitude of the oscillatory integral in the space \(C^k_0(\mathbb{R}^3)\). The uniform estimates of the absolute value of the oscillatory integrals by a value of order of \(C\tau^{-1}(\varphi)_2\) are sharp due to the fact, that there exists a germ \(Q\) at \(0\in\mathbb{R}^3\) with 4-jet \(\pm x^2+(y^2+ z^2)^2\) such that the germ \(Q\) is adjacent to the germ \(g\) at \(0\in\mathbb{R}^3\), but the oscillatory integral with phase \(Q\) decreases as \(\tau^{-1}\).