# zbMATH — the first resource for mathematics

Uniform estimates of oscillatory integrals with phase from the series $$\widetilde R_m$$. (English. Russian original) Zbl 0924.42013
Math. Notes 64, No. 3, 404-406 (1998); translation from Mat. Zametki 64, No. 3, 468-469 (1998).
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 $$\tau^{-1}$$.
##### MSC:
 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 58C99 Calculus on manifolds; nonlinear operators 58A20 Jets in global analysis
##### Keywords:
uniform estimates; oscillatory integrals; 4-jet; adjacent; germ
Full Text:
##### References:
 [1] V. N. Karpushkin, in:Trudy Sem. Petrovsk. [in Russian], Vol. 9, Izd. Moskov. Univ., Moscow (1983), pp. 1–39. [2] V. P. Maslov and M. V. Fedoryuk,Mat. Zametki [Math. Notes],30, No. 5, 763–768 (1981). [3] A. M. Chebotarev,Mat. Zametki [Math. Notes],34, No. 2, 273–280 (1983). [4] V. N. Karpushkin, in:Trudy Sem. Petrovsk. [in Russian], Vol. 10, Izd. Moskov. Univ., Moscow (1984), pp. 150–169. [5] V. I. Arnold, A. N. Varchenko, and S. M. Gusein-Zade,Singularities of Differentiable Maps [in Russian], Vol. 2, Nauka, Moscow (1984). [6] V. N. Karpushkin,Mat. Zametki [Math. Notes],56, No. 6, 131–133 (1994).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.