Cordial Volterra integral operators in spaces $L^{p}(0,T)$

Abstract

A necessary and sufficient condition for the core function $\varphi \in L^{1}(0,1)$ is established in order to define a bounded cordial Volterra integral operator $(V_{\varphi }u)(t)=\int _{0}^{t}t^{-1}\varphi (t^{-1}s)u(s)\,ds$ in a space $L^{p}(0,T)$ for a $p\in [1,\infty ]$. This condition implies the boundedness of $V_{\varphi }$ also in the Sobolev spaces $W^{m,p}(0,T)$, $m\geq 1$. The spectra of $V_{\varphi }$ in the spaces $L^{p}(0,T)$ and $W^{m,p}(0,T)$ are determined and the spectral properties of $V_{\varphi }$ are examined in $L^{p}(0,T)$.