Асимптотические разложения для одного класса сингулярных интегралов, возникающих в нелинейных волновых системах

Теоретическая и математическая физика. 2023. Т. 214. № 2. С. 179–197.

Dymov A. V.

    We find asymptotical expansions as $\nu \to 0$ for integrals of the form   $\int_{\mathbb{R}^d} F(x) / \big(\omega(x)^2 + \nu^2\big)\, dx$,
     where sufficiently smooth functions $F$ and  $\omega$ satisfy natural assumptions for their behaviour at infinity and all critical points of the function $\omega$ from the set $\{\omega(x) = 0\}$ are non-degenerate. 
         These asymptotics play a crucial role when analysing stochastic models for non-linear waves systems. 
    Our result generalizes that of \big[S. Kuksin, Russ. J.  Math. Phys.'2017\big] 
    where a similar asymptotics was found in a particular case when $\omega$ is a non-degenerate quadratic form of the signature $(d/2,d/2)$ with even $d$.

Keywords: Singular integrals, quadric, квадрика, сингулярный интеграл