Error estimate in the generalized Szegö theorem. (English) Zbl 0743.58032

Laptev, A.; Safarov, Yu.

Journ. Équ. Dériv. Partielles, St.-Jean-De-Monts 1991, No.XV, 7 p. (1991).

The authors announce the proof of a Szegö type result.
Let \(A\) be a positive selfadjoint elliptic pseudodifferential operator of order 1 on a smooth compact manifold \(M\) without boundary, \(n=\dim M\geq 2\). Then one can associate for any pseudodifferential operator \(B\) of order \(0\) and any continuous function \(f\) on \(\mathbb{R}\) the quantity: \[ \rho_ \lambda(f)=\hbox{Tr }f(\Pi_ \lambda B\Pi_ \lambda)\hbox{ where }\Pi_ \lambda\hbox{ is the spectral projector on }[0,\lambda[. \] If \(c_ 0(f)\lambda^ n\) corresponds the asymptotic behavior of \(\rho_ \lambda(f)\) as \(\lambda\) tends to \(\infty\) as was proved by V. Guillemin [Ann. Math. Stud. 91, 219-259 (1979; Zbl 0452.35093)] the authors obtain the following improvement:
Theorem. There exist \(r\) and a positive constant \(C\) such that for any \(f\) in \(C^ r(\mathbb{R})\) the following inequality holds: \[ | \rho_ \lambda(f)-c_ 0(f)\lambda^ n|\leq C(\lambda^{n-1}+1)\| f\|_{C^ r(K)} \] with \(K=[-\| B\|,\| B\|]\).
Moreover, “\(r=2\)” is obtained for particular \(B\).

Reviewer: B.Helffer (Paris)

MSC:

58J40	Pseudodifferential and Fourier integral operators on manifolds
58J50	Spectral problems; spectral geometry; scattering theory on manifolds
35P20	Asymptotic distributions of eigenvalues in context of PDEs

Keywords:

generalized Szegö theorem; elliptic pseudodifferential operator

Citations:

Zbl 0452.35093