Solutions of the Abel integral equation in Lebesgue spaces
type: Item
instance of: scholarly article
title: Solutions of the Abel integral equation in Lebesgue spaces
zbMATH Open document ID: 0576.45001
author: Loredana Biacino
published in: Ricerche di Matematica
publication date: 1984-01-01T00:00:00Z
review text: The fractional integration operator \(J_{\alpha}\), given by \[ J_{\alpha}u(x)=\frac{1}{\Gamma (\alpha)}\int^{x}_{0}(x-t)^{\alpha -1}u(t)dt,\quad 0<\alpha <1, \] is investigated for its mapping properties from Sobolev spaces \({}_ 0H^{s,p}(\Omega)\) to \({}_ 0H^{s+\alpha,p}\) and from spaces \(L^ p(\Omega)\) to \({}_ 0H^{\alpha,p}(\Omega)\), where \(\Omega =(0,1)\). The questions of interest are continuity, bijectivity, invertibility and continuity of the inverse operator. By \({}_ 0H^{s,p}\) is denoted the subspace of \(H^{s,p}\) consisting of functions u with (in the sense of trace) \(u(0)=u'(0)=\) \(=u^{(n-1)}(0)\) in case \(1/p+n-1<s<1/p+n\) (in particular \({}_ 0H^{s,p}=H^{s,p}\) if \(0\leq s<1/p)\).
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 26A33
zbMATH DE Number: 3921228
zbMATH Keywords: Solutions of the Abel integral equation in Lebesgue spaces
zbMATH Keywords: Abel
zbMATH Keywords: integral equations
zbMATH Keywords: fractional integration operator
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: continuity
zbMATH Keywords: bijectivity
zbMATH Keywords: invertibility
zbMATH Keywords: inverse operator
reviewed by: Rudolf Gorenflo
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3921228