Non-commutative residues for anisotropic pseudo-differential operators in \(\mathbb R^n\)
The authors define a noncommutative residue for the algebra of classical anisotropic pseudodifferential operators on \({\mathbb R}^n\) modulo regularizing operators. The pseudodifferential operators have symbols \(a(z)\in C^\infty({\mathbb R}^{2n})\) of class \(\Gamma^{\mu,M}({\mathbb R}^{2n})\), where \(\mu\in{\mathbb R}\) and \(M\) is a given \(2n\)-tuple of rational numbers \(>1\), if \(| \partial_z^\alpha(z)| \leq C_\alpha(1+| z| _M)^{\mu-\langle\alpha,M\rangle}\) uniformly for \(z\in{\mathbb R}^{2n}\) with \(| z| _M:=\sum_{i=1}^{2n}| z_i| ^{1/M_i}\) and \(\langle\alpha,M\rangle:= \sum_{i=1}^{2n}\alpha_i M_i\). The trace is defined by \(\text{ Res\; Op}(a)=\int_{{\mathbb S}^{2n-1}} a_{-| M| }(z)j^*\sigma\), where \(a_{-| M| }(z)\) is the term of quasi-homogeneity degree \(-| M| :=-(M_1+\cdots+M_{2n})\) in the expansion of \(a\), and \(j^*\sigma\) is a particular \((2n-1)\)-form on \({\mathbb S}^{2n-1}\). By means of the Weyl formula, it is shown that this trace coincides with the Dixmier trace on operators of order \(-| M| \). The proof highlights the differential-topology aspects of the problem and the relation between the uniqueness of the trace and the fact that the homology group \(H_{2n-1}({\mathbb R}^{2n}\setminus{0})\) has rank 1.
scientific article; zbMATH DE number 2005491
scientific article