Geodesic
Estimates for oscillatory integrals with convex phase
Izvestiya. Mathematics , Tome 70 (2006) no. 1, pp. 171-209.

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We consider methods for estimating one-dimensional oscillatory integrals with convex phase and amplitudes of bounded variation or Lipschitz class amplitudes. In particular, we improve the estimate for the Piercey integral with near-caustic parameter values, and also consider estimation methods for $n$-dimensional oscillatory integrals. 
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M. A. Chahkiev. Estimates for oscillatory integrals with convex phase. Izvestiya. Mathematics , Tome 70 (2006) no. 1, pp. 171-209. http://geodesic.mathdoc.fr/item/IM2_2006_70_1_a7/

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