On the measure of non-compactness and singular numbers for the Volterra integral operators (Q2709555)

The paper is devoted to the study of the Volterra integral operator NEWLINE\[NEWLINEK_{v}(f)(x)=v(x)\int^{x}_{0}k(x,y)f(y) dy\tag{1}NEWLINE\]NEWLINE with positive kernel \(k\) on the half-axis \(R_{+}=(0,\infty)\). Sufficient conditions are established for the operator \(K_{v}\) to be compact from the space \(L^{p}(R_{+})\) to \(L^{q}(R_{+})\) with \(1<p\leq q<\infty\). Two-sided estimates of the measure of non-compactness and Schatten-von Neumann norm are given for such operators \(K_{v}\) and for the multidimensional ball fractional integrals; for example, see Section 29.2, Note 25.15 in the book by \textit{S. G. Samko}, the reviewer and \textit{O. I. Marichev} [``Fractional integrals and derivatives. Theory and applications'' (1993; Zbl 0818.26003)]. Necessary and sufficient conditions are proved for \(K_{v}\) to be in a Schatten-von-Neumann ideal, and the estimate for \(K_{v}\) in this ideal is presented. Note that similar results for the Riemann-Liouville fractional operator \(R_{\alpha,v}\), being obtained from (1) when \(k(x,y)=(x-y)^{\alpha -1}\) \((\alpha >0)\), were proved by \textit{D. E. Edmunds} and \textit{V. D. Stepanov} [Math. Ann., 298, No. 1, 41-66 (1994; Zbl 0788.45013)] and by \textit{J. Newman} and \textit{M. Solomyak} [Integral Equations Oper. Theory, 20, No. 3, 335-349 (1994; Zbl 0817.47024)].