Complex powers of classical SG-pseudodifferential operators (Q2465923)

Denote by \(S ^{\mu ,m }(\mathbb R^{n} \times \mathbb R^{n})\) the class of matrix-valued symbols which satisfiy estimates of form \(| \partial _{\xi} ^{\alpha }\partial _{x} ^{\beta } a(x,\xi )| \leq C _{\alpha, \beta }[x] ^{m- |\beta |} [\xi ] ^{\mu -|\alpha |}\), \([x]=(1+|x| ^{2}) ^{1/2}\), \([\xi ]=(1+|\xi | ^{2}) ^{1/2}\). Functions in \(S ^{\mu ,m }(\mathbb R^{n} \times\mathbb R^{n})\) shall be called \(SG\)-symbols. A symbol is called a poly-homogeneous symbol if it is given by functions in \(S ^{\mu ,m }(\mathbb R^{n} \times \mathbb R^{n})\) which have both in the variable and in the covariable asymptotic expansions into components that are homogeneous of one step decreasing order. The authors study ellipticity and complex powers for pseudodifferential operators in both general and poly-homogeneous \(SG\)-symbol classes.