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 Trudy Mat. Inst. Steklova, 2020, Volume 310, Pages 161–175 (Mi tm4099)

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

A. V. Dymov

Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

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.

 Funding Agency Grant Number Russian Science Foundation 19-71-30012 This work is supported by the Russian Science Foundation under grant 19-71-30012.

DOI: https://doi.org/10.4213/tm4099

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English version:
Proceedings of the Steklov Institute of Mathematics, 2020, 310, 148–162

Bibliographic databases:

UDC: 517.3
Revised: December 21, 2019
Accepted: April 18, 2020

Citation: A. V. Dymov, “Asymptotic Estimates for Singular Integrals of Fractions Whose Denominators Contain Products of Block Quadratic Forms”, Selected issues of mathematics and mechanics, Collected papers. On the occasion of the 70th birthday of Academician Valery Vasil'evich Kozlov, Trudy Mat. Inst. Steklova, 310, Steklov Math. Inst., Moscow, 2020, 161–175; Proc. Steklov Inst. Math., 310 (2020), 148–162

Citation in format AMSBIB
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