Abstract

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.

中文翻译：

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分母包含块二次形式积的分数的奇异积分的渐近估计

摘要

对于形式为（（G（x）/ \ prod _ {\ alpha = 1} ^ {\ mathcal A}（Q_ \ alpha ）的比率的\（\ mathbb R ^ {dM} \）上的积分，获得非平凡的上限（x）+ i \ nu \ Gamma_ \ alpha（x））\）与\（\ nu \至0 \），其中\（Q_ \ alpha \）是由\（d \ times d \）组成的实数二次形式块\（\ Gamma_ \ alpha \）是远离零限制的实函数，而\（G \）是在无限远处具有足够快的衰减的函数。这种积分是在波动理论中产生的。特别是，它们在SB Kuksin和作者致力于严格研究四波相互作用的最新论文中起着关键作用。这些积分的分析简化为快速振荡积分的分析，该积分的相位函数在一部分变量中是二次方的，而在另一部分变量中是线性的，并且可能高度退化。