Pseudodifferential operators of Mellin type on Sobolev spaces
type: Item
instance of: scholarly article
title: Pseudodifferential operators of Mellin type on Sobolev spaces
zbMATH Open document ID: 0582.35117
DOI: 10.1007/BF01776852
author: Lorenza Diomeda
author: Benedetta Lisena
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1985-01-01T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 3931621
zbMATH Keywords: pseudodifferential operators
zbMATH Keywords: Mellin transform
zbMATH Keywords: meromorphic function
zbMATH Keywords: poles
zbMATH Keywords: growth conditions
zbMATH Keywords: Sobolev space
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3931621