Asymptotic Estimates for Singular Integrals of Fractions Whose Denominators Contain Products of Block Quadratic Forms

A nontrivial upper bound is obtained for integrals over $\mathbb R^{dM}$ of ratios of the form $G(x)/\prod _{\alpha =1}^{\mathcal A} (Q_\alpha (x)+i\nu \Gamma _\alpha (x))$ with $\nu \to 0$, where $Q_\alpha $ are real quadratic forms composed of $d\times d$ blocks, $\Gamma _\alpha $ are real functions bounded away from zero, and $G$ is a function with sufficiently fast decay at infinity. Such integrals arise in wave turbulence theory; in particular, they play a key role in the recent papers by S. B. Kuksin and the author devoted to the rigorous study of the four-wave interaction. The analysis of these integrals reduces to the analysis of rapidly oscillating integrals whose phase function is quadratic in a part of variables and linear in the other part of variables and may be highly degenerate.