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

A. V. Dymov

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected issues of mathematics and mechanics, Tome 310 (2020), pp. 161-175

Résumé

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.

Détail

Keywords: } \thanks {This work is supported by the Russian Science Foundation under grant 19-71-30012.