Residue traces for a pseudodifferential operator algebra on foliated manifolds (Q1402357)

Anisotropic (pseudo)differential operators are defined on products of Euclidean domains by assigning different weights to partial derivatives in different variables. For example, the heat operator \(\partial/{\partial t}+\Delta\) is an anisotropic operator on \(\mathbb{R}_+\times \mathbb{R}^n\), which moreover is elliptic in this sense. The author first notes that this definition extends to multi-foliated manifolds. Then he proves the existence and uniqueness (up to a multiplicative constant) of a trace functional on the algebra of anisotropic operators which vanishes on smoothing operators (the manifold is assumed connected). The proof is done along the lines of \textit{B. V.~Fedosov}, \textit{F.~ Golse}, \textit{E.~Leichtnam} and \textit{E.~Schrohe} [J. Funct. Anal. 142, 1--31 (1996; Zbl 0877.58005)], in particular the author does not assume continuity of the trace. This residue trace is then identified with Dixmier's trace on the domain of definition of the latter. This fact requires the Weyl asymptotic formula for the eigenvalues of quasi-elliptic anisotropic operators proved by \textit{L.~Rodino} and \textit{F.~Nicola} [Proc. Int. Conf. Geom., Anal. App., Varanasi, India, August 21-24, 2000, 47--61 (2001; Zbl 1031.35112)]. Note that the results are stated for scalar operators but they extend with minor changes to bundles (Remark 3.6(b)).