Geodesic
Uniform Estimates for Oscillatory Integrals and Volumes with the Varchenko Phase
Matematičeskie zametki, Tome 72 (2002) no. 5, pp. 688-692

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We obtain uniform estimates for oscillatory integrals and volumes with a perturbed Varchenko phase, which follow from the statements of general theorems (the case of an arbitrary perturbation). The estimates turn out to be nearly exact. We also derive exact estimates by an argument similar to the proofs of general theorems (the case of a partial perturbation). 
@article{MZM_2002_72_5_a6,
     author = {V. N. Karpushkin},
     title = {Uniform {Estimates} for {Oscillatory} {Integrals} and {Volumes} with the {Varchenko} {Phase}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {688--692},
     publisher = {mathdoc},
     volume = {72},
     number = {5},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2002_72_5_a6/}
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V. N. Karpushkin. Uniform Estimates for Oscillatory Integrals and Volumes with the Varchenko Phase. Matematičeskie zametki, Tome 72 (2002) no. 5, pp. 688-692. http://geodesic.mathdoc.fr/item/MZM_2002_72_5_a6/