## Arkiv för Matematik

• Ark. Mat.
• Volume 7, Number 3 (1967), 283-298.

### Asymptotic behavior of integrals connected with spectral functions for hypoelliptic operators

Jöran Friberg

#### 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 αj, 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.

#### Article information

Source
Ark. Mat., Volume 7, Number 3 (1967), 283-298.

Dates
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485893633

Digital Object Identifier
doi:10.1007/BF02591633

Mathematical Reviews number (MathSciNet)
MR221091

Zentralblatt MATH identifier
0154.35503

Rights

#### Citation

Friberg, Jöran. Asymptotic behavior of integrals connected with spectral functions for hypoelliptic operators. Ark. Mat. 7 (1967), no. 3, 283--298. doi:10.1007/BF02591633. https://projecteuclid.org/euclid.afm/1485893633

#### References

• Bergendal, G., Convergence ans summability of eigenfunction expansions connected with elliptic differential operators. Medd. Lunds Univ. Mat. Sem. 15, Lund 1959.
• Browder, F., The asymptotic distribution of eigenfunctions and eigenvalues for semielliptic differential operators. Proc. Nat. Acad. Sci., U.S.A., 43, 270–3 (1957).
• Friberg, J., Multi-quasielliptic polynomials. To appear in Ann. Scuola Norm. Sup. Pisa, 1967.
• Friberg, J., Principal parts and canonical factorizations of hypoelliptic polynomials in two variables. Rend. Sem. Mat. Univ. Padova, 37, 112–32 (1967).
• Gårding, L., On the asymptotic properties of the spectral function belonging to a selfadjoint semi-bounded extension of an elliptic differential operator. Kungl. Fysiogr. Sällsk. Lund Förh. 24, 1–18 (1954).
• Gorčakov, V. N., Asymptotic behavior of spectral functions for hypoelliptic operators of a certain class. Soviet Math. Dokl. 4, 1328–31 (1963).
• Grušin, V. V., Connections between local and global properties for solutions of hypoelliptic equations with constant coefficients. Mat. Sborn. 66 (1965).
• Mihailov, V. P., The behavior of certain classes of polynomials at infinity, Soviet Math. Dokl. 164, 1256–9 (1965).
• Nilsson, N., Asymptotic estimates for spectral functions connected with hypoelliptic differential operators. Ark. f. Mat. 5, 527–40 (1965).
• Pini, B., Osservazioni sulla ipoellitticità. Boll. Un. Mat. Ital. (3)18, 420–32 (1963).