Remarks on uniform combined estimates of oscillatory integrals with simple singularities

We consider the problem of constructing asymptotically exact (for ) uniform (with respect to parameters ) estimates for oscillatory integrals containing a large parameter . 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 , (in Arnold's classification) and use the special case of to discuss the possibility of generalizing our results.

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