Complex powers of elliptic pseudodifferential operators (Q1078782)

The purpose of this paper is the construction of complex powers of elliptic pseudo-differential operators A and the study of the analytic properties of the corresponding kernel \(k_ 3(x,y)\). The operator A is supposed to be written as: \(A=\sum^{N}_{k=0}O_ p(a_ k)\) where \(a_ 0\) is homogeneous in \(\xi\) of order \(m>0\) for \(| \xi | \geq 1\), \[ a_ k(x,t\xi)=t^{r_ k}(\ln t)^{w_ k} a_ k(x,\xi)\quad for\quad | \xi | =1,\quad t\geq c_ 0>1;\quad k=1,...,N-1, \] and for \(k=0,...,N\), \[ \sup_{x,\xi}(1+| \xi |^ 2)^{| \gamma | -t_ k}\quad | x^{\alpha} D_ x^{\beta} D^{\gamma}_{\xi} a_ k(x,\xi)| <+\infty,\quad m=t_ 0>...>t_ N. \] It is shown, in particular, that in this case, \(k_ s(x,x)\) cannot always be extended to \({\mathbb{C}}\).