Geodesic
Remarks on uniform combined estimates of oscillatory integrals
Izvestiya. Mathematics , Tome 72 (2008) no. 4, pp. 793-816

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We consider the problem of constructing asymptotically exact (for $\Omega\gg 1$) uniform (with respect to parameters $t=(t_1,t_2,\dots,t_m)$) estimates for oscillatory integrals containing a large parameter $\Omega$. We suggest a possible multidimensional analogue of Vinogradov's well-known estimate for one-dimensional integrals. Based on this suggestion, we estimate the integrals with singularities of type $A_k$, $D_4^{\pm}$ (in Arnold's classification) and use the special case of $D_5^\pm$ to discuss the possibility of generalizing our results. 
@article{IM2_2008_72_4_a7,
     author = {D. A. Popov},
     title = {Remarks on uniform combined estimates of oscillatory integrals},
     journal = {Izvestiya. Mathematics },
     pages = {793--816},
     publisher = {mathdoc},
     volume = {72},
     number = {4},
     year = {2008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2008_72_4_a7/}
}
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D. A. Popov. Remarks on uniform combined estimates of oscillatory integrals. Izvestiya. Mathematics , Tome 72 (2008) no. 4, pp. 793-816. http://geodesic.mathdoc.fr/item/IM2_2008_72_4_a7/