Pseudodifferential operators of Mellin type on Sobolev spaces (Q1069067)

The authors consider pseudodifferential operators of the form \[ A f(t)=(1/2\pi i)\int_{Re z=1/p}t^{-z} a(t,z) \tilde f(z) dz, \] where \(1<p<\infty\) and \(\tilde f(\)z)\(=\int^{\infty}_{0}t^{z-1} f(t) dt\) is the Mellin transform of f. a(t,z) is a \(C^{\infty}\) function of \(t\geq 0\) and a meromorphic function of z in a strip \(c<Re z<d\) with a finite number of poles and suitable growth conditions when \(| Im z| \to \infty\). The continuity properties of A are studied on the Sobolev space \(W^{p,k}(0,\infty)\) of \(L^ p\)-functions on (0,\(\infty)\) with k derivatives in \(L^ p(0,\infty)\). In fact A is continuous from a subspace of \(W^{p,k}(0,\infty)\) defined by compatibility conditions to \(W^{p,k}(0,\infty)\). A notion of ellipticity is introduced and results of Fredholm type are proved for the operators A. The difficulty of the study comes from the presence of poles for a(.,z).