On the asymptotic expansion of some integrals (Q788226)

This note deals with integrals of the form \[ I(s,f):=\int_{{\mathbb{R}}^ n}g(x^{\alpha}/s)x^{\beta}\log^{\gamma}xf(x)dx \] where \(g\in S({\mathbb{R}}^ n)\) (the Schwartz space), \(f\in C_ 0^{\infty}({\mathbb{R}}^ n)\), \(s>0\), and \(\alpha,\beta \in {\mathbb{R}}^ n\) with \(\alpha_ i>0\), \(\beta_ i\geq 0\), 1\(\leq i\leq n\), \(\gamma \in {\mathbb{Z}}^ n_+\). The asymptotic expansion of I(s,f) as \(s\to 0\) is derived by induction on n using a two-variable asymptotic expansion in the induction step. As a special case one obtains a recent result of \textit{D. Barlet} [Invent. Math. 68, 129-174 (1982; Zbl 0508.32003)].