Solutions of the Abel integral equation in Lebesgue spaces
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 \(0\leq s<1/p)\).
scientific article; zbMATH DE number 3921228