Pseudodifferential operators of Mellin type on Sobolev spaces
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).
scientific article; zbMATH DE number 3931621
scientific article