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. 
@article{IM2_2006_70_1_a7,
     author = {M. A. Chahkiev},
     title = {Estimates for oscillatory integrals with convex phase},
     journal = {Izvestiya. Mathematics },
     pages = {171--209},
     publisher = {mathdoc},
     volume = {70},
     number = {1},
     year = {2006},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2006_70_1_a7/}
}
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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/