Fischer, Véronique

Real Trace Expansions

Doc. Math. 24, 2159-2202 (2019)
DOI: 10.25537/dm.2019v24.2159-2202
Communicated by Clotilde Fermanian Kammerer


In this paper, we investigate trace expansions of operators of the form \(A\eta(t\mathcal{L})\) where \(\eta:\mathbb{R}\rightarrow\mathbb{C}\) is a Schwartz function, \(A\) and \(\mathcal L\) are classical pseudo-differential operators on a compact manifold \(M\) with \(\mathcal L\) elliptic. In particular, we show that, under certain hypotheses, this trace admits an expansion in powers of \(t\rightarrow 0^+\). We also relate the constant coefficient to the non-commutative residue and the canonical trace of \(A\). Our main tool is the continuous inclusion of the functional calculus of \(\mathcal{L}\) into the pseudo-differential calculus whose proof relies on the Helffer-Sjöstrand formula.

Mathematics Subject Classification

58J40, 58J42


pseudodifferential operators on manifolds, non-commutative residues, canonical trace


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Fischer, Véronique
Department of Mathematical Sciences, University of Bath, BA2 7AY Bath, UK