Remarks on uniform combined estimates of oscillatory integrals with simple singularities

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.