Sub-elliptic functional calculus and non-commutative residue on Heisenberg manifolds (Q2738866)

The author's results on subelliptic functional calculus and non-commutative residue on Heisenberg manifolds are reported. -- A Heisenberg manifold is a manifold \(M\) with a hyperplane sub-fibre \(H\subset TM\). Let \(X_0,X_1,\dots,X_d\) be tangent vector fields such that \(H\) is spanned by \(X_1,\dots,X_d\), \(\xi_0,\xi_1, \dots,\xi_d\) the corresponding local coordinate. The anisotropic dilatations are defined by \(\lambda\cdot \xi=(\lambda^2 \xi_0, \lambda \xi_1, \dots, \lambda\xi_d)\). The space \(S^m(U\times \mathbb{R}^{d+1})\) of symbols of order \(m\in\mathbb{C}\) is the space of smooth functions on \(U\times \mathbb{R}^{d+1}\) such that \(f\sim\sum_{j\geq o}f_{m-j}\), where \(f_k(x,\xi)\) is homogeneous of degree \(k\) in \(\xi\) with respect to the anisotropic dilatations. \(A\Psi_HDO\) of degree \(m\) on \(U\) is an operator \(P=f(x, \sigma(x,D))+R\), \(f\in S^m(U\times \mathbb{R}^{d+1})\) and \(R\) is the regularization operator. A sub-Laplacian is an order \(2\Psi_HDO\) with the principal symbol \(\sum^d_{j=1} \xi^2_J+ \lambda(x) \xi_0\). It is known that the kernel of integral degree \(\Psi_HDO\) has asymptotic expansion of the form \(p_x(\lambda^{-1})-c_p(x) \log \lambda+ O(1)\) near the diagonal, and \(c_p(x)\) defines a density of \(M\) [\textit{R. Beals} and \textit{P. Greiner}, Calculus on Heisenberg Manifolds, Ann. Math. Studies, 119, Princeton (1988; Zbl 0654.58033)]. The author also introduces a \(\text{Hol}^p (\Lambda)\)-family \(\Psi_H^{p,m}(U)\) of \(\Psi_H (DO)\), where \(\text{Hol}^p (\Lambda)\) is the space of holomorphic functions on a pseudo-cone \(\Lambda\) such that \(|h(\lambda) |\leq C_{\Lambda'} (1+\lambda \mid)^p\) for any pseudo-cone \(\Lambda'\Subset \Lambda\). By using this notion, existence of ray \(L\subset \{{\mathfrak R}\lambda <0\}\) such that sub-elliptic sub-Laplacian \(\Delta-\lambda\), \(\lambda\in L\) has the resolvent (Theorem 1). Existence of complex powers of \(\Delta\) when it is invertible and existence of 1-parameter holomorphic family \(\Delta^s\) of \(\Psi_H(DO)\) are also shown (Theorem 2). If \({\mathfrak R}\text{ord} P<- (d+2)\), by using its kernel, \(\text{Trace} P= \int_Mk_P(x,x)\) is defined, and the map \(P\mapsto\text{Trace}(P)\) allows analytic continuation on the space of non integral order \(\Psi_HDO\). If \(P\in \Psi^\mathbb{Z}_H\) is contained in a holomorphic family \((P_z)\) of \(\Psi_HDO\) such that \(P_0=P\), \(\text{ord} P_z= z+\text{ord} P\), then \(\text{Trace} P_z\) has a simple pole at \(z=0\) with the residue \(-\int_Mc_p(x)\) (Theorem 3). By this Theorem, non-commutative residue is defined by \(\text{Res} P=\int_m c_p(x)\). By definition \(\text{Res} P=2\text{res}_{s=0} \text{Trace} P \Delta^{-s}\) for any integral degree \(P\). It is also stated Res is the essentially unique trace on \(\Psi^\mathbb{Z}_H(M)/ \Psi^{-\infty}(M)\) (Theorem 4).