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RESEARCH PRODUCT

Complex powers of elliptic pseudodifferential operators

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The aim of this paper is the construction of complex powers of elliptic pseudodifferential operators and the study of the analytic properties of the corresponding kernels kS (x,y). For x=y, the case of principal interest, the domain of holomorphy and the singularities of kS (x,x) are shown to depend on the asymptotic expansion of the symbol. For classical symbols, kS (x,x) is known to be meromorphic on ℂ with simple poles in a set of equidistant points on the real axis. In the more general cases considered here, the singularities may be distributed over a half plane and kS (x,x) can not always be extended to337-2. An example is given where kS (x,x) has a vertical line as natural boundary.