Asymptotic behaviour of fundamental solutions of elliptic operators with order higher than two (Q1288000)

Let \(F(D)= \sum_{|\alpha|= 2m} F_\alpha D^\alpha\) be a homogeneous elliptic operator of order \(2m\) \((m>2)\) with constant coefficients on \(\mathbb{R}^n\), and let the fundamental solution \(E(x)\) to the operator \(F(D)- I\) be bounded as \(x\to\infty\). The author describes the asymptotic behaviour of \(E(x)\) in terms of coefficients \(F_\alpha\) when the energy surface \(S= \{F(\xi)= 1\}\) has points where its total curvature vanishes. The proofs are based on some estimates for oscillatory integrals obtained by \textit{A. N. Varčenko} [Funct. Anal. Appl. 10, 175-196 (1976; Zbl 0351.32011)].