Geodesic
Asymptotic Estimates for Singular Integrals of Fractions Whose Denominators Contain Products of Block Quadratic Forms
Informatics and Automation, Selected issues of mathematics and mechanics, Tome 310 (2020), pp. 161-175

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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.
Keywords: } \thanks {This work is supported by the Russian Science Foundation under grant 19-71-30012. 
A. V. Dymov. Asymptotic Estimates for Singular Integrals of Fractions Whose Denominators Contain Products of Block Quadratic Forms. Informatics and Automation, Selected issues of mathematics and mechanics, Tome 310 (2020), pp. 161-175. http://geodesic.mathdoc.fr/item/TRSPY_2020_310_a11/
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     title = {Asymptotic {Estimates} for {Singular} {Integrals} of {Fractions} {Whose} {Denominators} {Contain} {Products} of {Block} {Quadratic} {Forms}},
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[1] Buckmaster T., Germain P., Hani Z., Shatah J., Onset of the wave turbulence description of the longtime behavior of the nonlinear Schrödinger equation, E-print, 2019, arXiv: 1907.03667 [math.AP]

[2] Dimassi M., Sjöstrand J., Spectral asymptotics in the semi-classical limit, LMS Lect. Note Ser., 268, Cambridge Univ. Press, Cambridge, 1999 | MR | Zbl

[3] Dymov A., Kuksin S., Formal expansions in stochastic model for wave turbulence. 1: Kinetic limit, E-print, 2019, arXiv: 1907.04531 [math-ph]

[4] Dymov A., Kuksin S., Formal expansions in stochastic model for wave turbulence. 2: Method of diagram decomposition, E-print, 2019, arXiv: ; Dymov A., Kuksin S., On the Zakharov–L'vov stochastic model for wave turbulence, E-print, 2019, arXiv: 1907.02279 [math-ph]1907.05044 [math-ph]

[5] A. V. Dymov and S. B. Kuksin, “On the Zakharov–L'vov stochastic model for wave turbulence”, Dokl. Math., 101:2 (2020), 102–109 | DOI

[6] Hörmander L., The analysis of linear partial differential operators, v. I, Grundl. Math. Wiss., 256, Distribution theory and Fourier analysis, Springer, Berlin, 1983 | MR | Zbl

[7] Kuksin S., “Asymptotic expansions for some integrals of quotients with degenerated divisors”, Russ. J. Math. Phys., 24:4 (2017), 476–487 | DOI | MR | Zbl

[8] Kuksin S., “Asymptotic properties of integrals of quotients when the numerator oscillates and the denominator degenerates”, J. Math. Phys. Anal. Geom., 14:4 (2018), 510–518 | MR | Zbl

[9] Lukkarinen J., Spohn H., “Weakly nonlinear Schrödinger equation with random initial data”, Invent. math., 183:1 (2011), 79–188 | DOI | MR | Zbl

[10] Nazarenko S., Wave turbulence, Lect. Notes Phys., 825, Springer, Berlin, 2011 | DOI | MR | Zbl

[11] Newell A.C., Rumpf B., “Wave turbulence”, Annu. Rev. Fluid Mech., 43 (2011), 59–78 | DOI | MR | Zbl

[12] V. E. Zakharov and V. S. L'vov, “Statistical description of nonlinear wave fields”, Radiophys. Quantum Electron., 18:10 (1975), 1084–1097 | DOI | MR

[13] Zakharov V.E., L'vov V.S., Falkovich G., Kolmogorov spectra of turbulence, v. I, Wave turbulence, Springer, Berlin, 1992 | MR | Zbl