Asymptotic behavior of integrals connected with spectral functions for hypoelliptic operators

Abstract

In the first part of this paper are considered real polynomials P(ζ), ζ∈Rⁿ, complete and nondegenerate in the sense that there is a set of (even) multi-indices α^j, j=1,..., N, such that, for |ζ|> K, ζ real, $cP(\xi ) \leqslant \sum {\xi ^{\alpha j} } \leqslant CP(\xi ).$ (See V. P. Mihailov, Soviet Math. Dokl. 164 (1965), MR 32: 6047).

It is then proved by an explicit computation, for every given even multi-index γ, that there are a real number θ>0 and an integer r, 0≤r< n, depending only on γ and {α¹}, and such that $\int {\xi ^\gamma } \exp \{ - tP(\xi )\} d\xi - K\gamma (P)t^{ - \theta } \left| {\log t} \right|^\gamma (1 + o(1))$ as t→+0. A Tauberian argument then leads to an asymptotic estimate of the integral $e_0^{(\beta ,\beta )} (\lambda ,0) = \int {{}_{P(\xi \leqslant \lambda )}\xi ^{2\beta } d\xi ,} $ , where e ${}_{0}^{(β, β)}$ is a derivative of a certain spectral function. Less explicit results for a larger class of polynomials were given by N. Nilsson, Ark. f. Mat. 5 (1965). In the second part of the paper, the explicit computations are extended to the larger class considered by Nilsson but under the restriction n=2.