On the singular numbers of certain Volterra integral operators
The authors consider an integral operator \(K\) of the form \[ (Kf) (x) = v(x) \int^x_0 k(x,y) u(y)f(y)dy, \quad x \geq 0,\;f \in L^2 (0, \infty), \] where \(u\), \(v\) are real-valued functions satisfying certain local integrability conditions, \(k\) is a nonnegative function with the property that there is a constant \(D \geq 1\) such that \[ D^{-1} \bigl( k(x,y) + k(y,z) \bigr) \leq k(x,z) \leq D \bigl( k(x,y) + k(y,z) \bigr),\;x > y > z \geq 0. \] Criteria for \(K\) to be bounded, compact, or to belong to the Schatten-von Neumann ideals are given. In particular, when \(2 \leq p < \infty\), it is shown that the Schatten \(p\)-norm of such an operator can be estimated by constant multiplies of an integral expression which is exactly the well-known formula for the Hilbert-Schmidt norm when \(p = 2\).
scientific article; zbMATH DE number 847470