Asymptotics for a class of singular integrals of quotients with highly degenerate denominators

In rigorous study of stochastic models for the wave turbulence theory and R. Peierls’s kinetic theory for the thermal conductivity in solids, analysis of integrals of the form ${\int }_{\mathcal{M}}\frac{F{\omega }_{\mathcal{M}}}{{\mathrm{\Omega }}^2+{\nu }^2{\mathrm{\Gamma }}^2}$ and ${\int }_{\mathcal{M}}\frac{F\cos ({\nu }^{-1}\mathrm{\Omega }){\omega }_{\mathcal{M}}}{{\mathrm{\Omega }}^2+{\nu }^2{\mathrm{\Gamma }}^2}$ plays a crucial role, where $\nu >0$ is a small parameter, $\mathcal{M}$ is a closed Riemannian manifold with volume form ${\omega }_{\mathcal{M}}$, and the functions $\mathrm{\Gamma }>0$, $F$, $\mathrm{\Omega }$ are sufficiently smooth. We investigate the asymptotic behavior of the integrals in the limit $\nu \to 0$. This work continues studies [Kuksin’ 17, Dymov’ 23], in which the authors considered similar integrals for the case $\mathcal{M}={\mathbb{R}}^d$ when the function $\mathrm{\Omega }$ is Morse. We significantly weaken the latter assumption, which played an important role in the aforementioned works. This makes the obtained results applicable to the problem of rigorous justification of R. Peierls’s kinetic theory.