Geodesic
Asymptotic properties of integrals of quotients when the numerator oscillates and the denominator degenerates
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018)

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We study asymptotic expansion as $\nu\to0$ for integrals over ${ \mathbb{R} }^{2d}=\{(x,y)\}$ of quotients of the form $F(x,y) \cos(\lambda x\cdot y) \big/ \big( (x\cdot y)^2+\nu^2\big)$, where $\lambda\ge 0$ and $F$ decays at infinity sufficiently fast. Integrals of this kind appear in the theory of wave turbulence. 
@article{JMAG_2018_14_a3,
     author = {Sergei Kuksin},
     title = {Asymptotic properties of integrals of quotients when the numerator oscillates and the denominator degenerates},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     publisher = {mathdoc},
     volume = {14},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2018_14_a3/}
}
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Sergei Kuksin. Asymptotic properties of integrals of quotients when the numerator oscillates and the denominator degenerates. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018). http://geodesic.mathdoc.fr/item/JMAG_2018_14_a3/